ベル数の定義
ベル数の定義
区別の出来る\(n\)個の玉を区別の出来ない\(k\)個の箱に空箱ありで分ける方法の数を\(B\left(n,k\right)\)で表しベル数という。
空箱なしの場合の数は\(S_{2}\left(n,k\right)\)なので、箱が区別出来ないときの空箱ありなしの関係より\(B\left(n,k\right)\)は、
\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =\sum_{m=0}^{k}\frac{m^{n}}{m!}\sum_{j=0}^{k-m}\frac{\left(-1\right)^{j}}{j!} \end{align*} となる。
また、区別の出来る\(n\)個の玉を箱に分ける方法の数を\(B\left(n\right)\)で表し、
\begin{align*} B\left(n\right) & =B\left(n,n\right)\\ & =\sum_{j=0}^{n}S_{2}\left(n,j\right)\\ & =\sum_{m=0}^{n}\frac{m^{n}}{m!}\sum_{j=0}^{n-m}\frac{\left(-1\right)^{j}}{j!} \end{align*} となる。
ベル数は\(n<0\lor k<0\)のとき、\(B\left(n,k\right)=0\)となる。
区別の出来る\(n\)個の玉を区別の出来ない\(k\)個の箱に空箱ありで分ける方法の数を\(B\left(n,k\right)\)で表しベル数という。
空箱なしの場合の数は\(S_{2}\left(n,k\right)\)なので、箱が区別出来ないときの空箱ありなしの関係より\(B\left(n,k\right)\)は、
\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =\sum_{m=0}^{k}\frac{m^{n}}{m!}\sum_{j=0}^{k-m}\frac{\left(-1\right)^{j}}{j!} \end{align*} となる。
また、区別の出来る\(n\)個の玉を箱に分ける方法の数を\(B\left(n\right)\)で表し、
\begin{align*} B\left(n\right) & =B\left(n,n\right)\\ & =\sum_{j=0}^{n}S_{2}\left(n,j\right)\\ & =\sum_{m=0}^{n}\frac{m^{n}}{m!}\sum_{j=0}^{n-m}\frac{\left(-1\right)^{j}}{j!} \end{align*} となる。
ベル数は\(n<0\lor k<0\)のとき、\(B\left(n,k\right)=0\)となる。
\(B\left(3,1\right)\)は\(\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} \)の1通り
\(B\left(3,2\right)\)は\(\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} \),\(\left\{ \left\{ a\right\} ,\left\{ b,c\right\} \right\} ,\left\{ \left\{ b\right\} ,\left\{ c,a\right\} \right\} ,\left\{ \left\{ c\right\} ,\left\{ a,b\right\} \right\} \)の4通り
\(B\left(3,3\right)\)は\(\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} ,\left\{ \left\{ a\right\} ,\left\{ b,c\right\} \right\} ,\left\{ \left\{ b\right\} ,\left\{ c,a\right\} \right\} ,\left\{ \left\{ c\right\} ,\left\{ a,b\right\} \right\} ,\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} \)の5通り
\(B\left(3,2\right)\)は\(\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} \),\(\left\{ \left\{ a\right\} ,\left\{ b,c\right\} \right\} ,\left\{ \left\{ b\right\} ,\left\{ c,a\right\} \right\} ,\left\{ \left\{ c\right\} ,\left\{ a,b\right\} \right\} \)の4通り
\(B\left(3,3\right)\)は\(\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} ,\left\{ \left\{ a\right\} ,\left\{ b,c\right\} \right\} ,\left\{ \left\{ b\right\} ,\left\{ c,a\right\} \right\} ,\left\{ \left\{ c\right\} ,\left\{ a,b\right\} \right\} ,\left\{ \left\{ a\right\} ,\left\{ b\right\} ,\left\{ c\right\} \right\} \)の5通り
ベル数\(B\left(n,k\right)\)一覧
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 2 & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ \hline 3 & 0 & 1 & 4 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5\\ \hline 4 & 0 & 1 & 8 & 14 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15\\ \hline 5 & 0 & 1 & 16 & 41 & 51 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52\\ \hline 6 & 0 & 1 & 32 & 122 & 187 & 202 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203\\ \hline 7 & 0 & 1 & 64 & 365 & 715 & 855 & 876 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877\\ \hline 8 & 0 & 1 & 128 & 1,094 & 2,795 & 3,845 & 4,111 & 4,139 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140\\ \hline 9 & 0 & 1 & 256 & 3,281 & 11,051 & 18,002 & 20,648 & 21,110 & 21,146 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147\\ \hline 10 & 0 & 1 & 512 & 9,842 & 43,947 & 86,472 & 109,299 & 115,179 & 115,929 & 115,974 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975\\ \hline 11 & 0 & 1 & 1,024 & 29,525 & 175,275 & 422,005 & 601,492 & 665,479 & 677,359 & 678,514 & 678,569 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570\\ \hline 12 & 0 & 1 & 2,048 & 88,574 & 700,075 & 2,079,475 & 3,403,127 & 4,030,523 & 4,189,550 & 4,211,825 & 4,213,530 & 4,213,596 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597\\ \hline 13 & 0 & 1 & 4,096 & 265,721 & 2,798,251 & 10,306,752 & 19,628,064 & 25,343,488 & 27,243,100 & 27,602,602 & 27,641,927 & 27,644,358 & 27,644,436 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437\\ \hline 14 & 0 & 1 & 8,192 & 797,162 & 11,188,907 & 51,263,942 & 114,700,315 & 164,029,595 & 184,941,915 & 190,077,045 & 190,829,797 & 190,895,863 & 190,899,230 & 190,899,321 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322\\ \hline 15 & 0 & 1 & 16,384 & 2,391,485 & 44,747,435 & 255,514,355 & 676,207,628 & 1,084,948,961 & 1,301,576,801 & 1,368,705,291 & 1,381,367,941 & 1,382,847,419 & 1,382,953,889 & 1,382,958,439 & 1,382,958,544 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545\\ \hline 16 & 0 & 1 & 32,768 & 7,174,454 & 178,973,355 & 1,275,163,905 & 4,010,090,463 & 7,291,973,067 & 9,433,737,120 & 10,254,521,370 & 10,448,276,360 & 10,477,213,268 & 10,479,970,386 & 10,480,136,006 & 10,480,142,026 & 10,480,142,146 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147\\ \hline 17 & 0 & 1 & 65,536 & 21,523,361 & 715,860,651 & 6,368,612,302 & 23,874,362,200 & 49,582,466,986 & 69,998,462,014 & 79,527,284,317 & 82,285,618,467 & 82,797,679,445 & 82,859,701,769 & 82,864,611,947 & 82,864,861,847 & 82,864,869,667 & 82,864,869,803 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804\\ \hline 18 & 0 & 1 & 131,072 & 64,570,082 & 2,863,377,067 & 31,821,472,612 & 142,508,723,651 & 339,971,207,051 & 529,007,272,061 & 635,182,667,816 & 672,294,831,619 & 680,685,836,527 & 681,942,165,393 & 682,068,020,031 & 682,076,428,809 & 682,076,796,009 & 682,076,806,005 & 682,076,806,158 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159\\ \hline 19 & 0 & 1 & 262,144 & 193,710,245 & 11,453,377,195 & 159,042,661,905 & 852,124,263,684 & 2,345,048,898,523 & 4,054,799,902,003 & 5,199,414,528,808 & 5,676,711,562,593 & 5,806,124,780,384 & 5,829,591,731,684 & 5,832,484,170,844 & 5,832,727,748,374 & 5,832,741,665,152 & 5,832,742,192,288 & 5,832,742,204,885 & 5,832,742,205,056 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057\\ \hline 20 & 0 & 1 & 524,288 & 581,130,734 & 45,813,246,635 & 795,019,337,135 & 5,101,098,232,519 & 16,244,652,278,171 & 31,415,584,940,850 & 43,426,867,585,575 & 49,344,452,550,230 & 51,245,294,979,716 & 51,656,311,613,107 & 51,717,380,273,487 & 51,723,682,798,067 & 51,724,135,127,267 & 51,724,157,478,221 & 51,724,158,219,506 & 51,724,158,235,181 & 51,724,158,235,371 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372\\ \hline 21 & 0 & 1 & 1,048,576 & 1,743,392,201 & 183,252,462,251 & 3,974,515,030,652 & 30,560,194,493,456 & 112,871,151,708,404 & 245,382,167,055,488 & 368,654,643,520,692 & 439,841,775,811,967 & 466,668,627,500,968 & 473,501,669,531,146 & 474,706,578,749,477 & 474,855,882,753,977 & 474,868,970,216,557 & 474,869,780,161,021 & 474,869,815,113,820 & 474,869,816,137,255 & 474,869,816,156,540 & 474,869,816,156,750 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751\\ \hline 22 & 0 & 1 & 2,097,152 & 5,230,176,602 & 733,008,800,427 & 19,870,830,712,482 & 183,176,170,057,707 & 785,938,550,025,147 & 1,928,337,630,016,767 & 3,170,300,933,550,687 & 4,005,444,732,928,641 & 4,371,727,233,798,927 & 4,480,550,589,850,064 & 4,503,047,451,718,545 & 4,506,342,616,999,876 & 4,506,688,232,943,076 & 4,506,714,279,517,080 & 4,506,715,683,659,127 & 4,506,715,737,033,756 & 4,506,715,738,423,606 & 4,506,715,738,447,091 & 4,506,715,738,447,322 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323\\ \hline 23 & 0 & 1 & 4,194,304 & 15,690,529,805 & 2,932,033,104,555 & 99,348,921,288,655 & 1,098,318,779,272,060 & 5,480,960,778,389,365 & 15,222,915,798,289,765 & 27,542,984,610,086,665 & 37,136,385,907,400,125 & 42,000,637,216,351,225 & 43,672,799,989,835,155 & 44,074,082,550,176,545 & 44,142,711,725,983,660 & 44,151,191,130,412,991 & 44,151,953,491,540,255 & 44,152,003,408,529,058 & 44,152,005,773,414,427 & 44,152,005,853,196,206 & 44,152,005,855,055,756 & 44,152,005,855,084,092 & 44,152,005,855,084,345 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346\\ \hline 24 & 0 & 1 & 8,388,608 & 47,071,589,414 & 11,728,128,223,915 & 496,728,911,719,165 & 6,586,964,947,803,695 & 38,264,428,799,608,235 & 120,582,710,957,928,740 & 241,205,285,284,001,240 & 349,459,367,068,932,740 & 412,559,532,764,708,300 & 437,489,737,355,466,560 & 444,378,573,413,388,560 & 445,740,664,435,029,560 & 445,936,484,677,276,640 & 445,957,161,859,742,195 & 445,958,772,809,679,110 & 445,958,865,294,604,555 & 445,958,869,175,343,725 & 445,958,869,292,316,504 & 445,958,869,294,771,110 & 445,958,869,294,805,012 & 445,958,869,294,805,288 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289\\ \hline 25 & 0 & 1 & 16,777,216 & 141,214,768,241 & 46,912,504,507,051 & 2,483,597,478,617,802 & 39,510,014,478,620,232 & 267,342,497,477,336,542 & 957,566,218,595,705,122 & 2,125,487,669,688,678,127 & 3,328,651,061,864,065,627 & 4,131,006,966,302,528,287 & 4,493,269,587,087,402,967 & 4,607,754,660,431,147,227 & 4,633,712,770,792,043,227 & 4,638,012,165,447,390,427 & 4,638,538,820,609,086,387 & 4,638,586,883,940,479,497 & 4,638,590,159,619,074,422 & 4,638,590,325,838,044,097 & 4,638,590,332,058,238,847 & 4,638,590,332,226,758,352 & 4,638,590,332,229,958,802 & 4,638,590,332,229,999,052 & 4,638,590,332,229,999,352 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353\\ \hline 26 & 0 & 1 & 33,554,432 & 423,644,304,722 & 187,650,001,250,987 & 12,417,846,161,543,552 & 237,013,033,135,668,883 & 1,868,866,831,126,685,483 & 7,618,489,083,072,350,433 & 18,820,005,864,027,476,058 & 32,019,561,236,874,324,063 & 42,048,639,577,872,800,823 & 47,198,146,931,729,759,643 & 49,048,715,505,983,309,703 & 49,526,614,124,379,597,963 & 49,617,063,154,570,701,963 & 49,629,789,031,813,184,523 & 49,631,132,763,608,563,353 & 49,631,239,789,154,665,113 & 49,631,246,222,993,683,863 & 49,631,246,513,616,548,538 & 49,631,246,523,375,652,893 & 49,631,246,523,614,582,298 & 49,631,246,523,618,708,498 & 49,631,246,523,618,755,948 & 49,631,246,523,618,756,273 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274\\ \hline 27 & 0 & 1 & 67,108,864 & 1,270,932,914,165 & 750,599,971,449,515 & 62,088,807,129,858,605 & 1,421,890,125,134,903,156 & 13,069,461,898,046,144,687 & 60,698,293,711,602,480,887 & 167,261,566,992,144,276,462 & 310,458,637,501,567,882,137 & 433,978,054,625,397,974,502 & 505,801,221,212,679,957,102 & 535,008,120,031,833,066,702 & 543,549,269,263,634,652,402 & 545,383,903,334,897,500,662 & 545,677,966,400,968,325,622 & 545,713,535,718,732,248,292 & 545,716,805,910,357,458,802 & 545,717,035,178,844,916,812 & 545,717,047,425,141,229,062 & 545,717,047,920,705,285,192 & 545,717,047,935,720,836,457 & 545,717,047,936,054,668,462 & 545,717,047,936,059,933,462 & 545,717,047,936,059,989,037 & 545,717,047,936,059,989,388 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389\\ \hline 28 & 0 & 1 & 134,217,728 & 3,812,798,742,494 & 3,002,399,818,689,195 & 310,442,764,649,269,995 & 8,530,588,879,837,946,391 & 91,423,392,608,221,681,659 & 484,101,618,889,583,612,790 & 1,490,799,910,228,016,109,165 & 3,029,333,888,602,793,961,490 & 4,531,244,547,474,348,583,180 & 5,516,641,963,645,562,466,745 & 5,968,154,814,881,834,874,145 & 6,116,937,802,946,210,183,545 & 6,152,998,463,246,954,493,145 & 6,159,538,106,375,350,540,765 & 6,160,436,847,843,408,051,115 & 6,160,531,280,610,425,762,965 & 6,160,538,906,903,312,675,665 & 6,160,539,381,097,726,378,675 & 6,160,539,403,750,867,869,655 & 6,160,539,404,576,774,053,615 & 6,160,539,404,599,467,740,995 & 6,160,539,404,599,927,933,000 & 6,160,539,404,599,934,587,375 & 6,160,539,404,599,934,652,076 & 6,160,539,404,599,934,652,454 & 6,160,539,404,599,934,652,455 & 6,160,539,404,599,934,652,455 & 6,160,539,404,599,934,652,455\\ \hline 29 & 0 & 1 & 268,435,456 & 11,438,396,227,481 & 12,009,599,140,539,051 & 1,552,210,010,313,389,752 & 51,180,527,066,276,028,928 & 639,650,299,280,150,852,200 & 3,863,968,913,259,430,036,516 & 13,316,931,761,586,684,435,022 & 29,708,969,836,672,895,454,647 & 47,768,521,062,634,774,145,562 & 61,095,200,715,560,895,370,032 & 67,950,265,197,803,650,549,797 & 70,484,739,881,941,177,288,797 & 71,174,432,774,516,717,242,197 & 71,315,127,724,871,798,313,717 & 71,336,945,972,957,172,037,287 & 71,339,544,504,231,548,360,937 & 71,339,783,836,563,417,414,087 & 71,339,800,946,744,578,386,987 & 71,339,801,896,654,963,400,577 & 71,339,801,937,478,040,938,677 & 71,339,801,938,825,901,932,377 & 71,339,801,938,859,640,227,877 & 71,339,801,938,860,266,779,257 & 71,339,801,938,860,275,115,858 & 71,339,801,938,860,275,190,765 & 71,339,801,938,860,275,191,171 & 71,339,801,938,860,275,191,172 & 71,339,801,938,860,275,191,172\\ \hline 30 & 0 & 1 & 536,870,912 & 34,315,188,682,442 & 48,038,396,293,720,747 & 7,761,038,612,902,285,822 & 307,071,141,359,850,971,579 & 4,475,987,863,912,937,373,659 & 30,859,006,547,961,045,671,459 & 119,159,990,796,885,614,442,329 & 292,533,334,396,074,979,037,085 & 507,580,435,956,741,855,656,775 & 685,560,143,017,817,189,041,330 & 788,002,660,939,899,127,602,745 & 830,340,371,000,067,257,128,510 & 843,220,239,072,837,883,168,510 & 846,161,051,171,094,720,266,230 & 846,672,656,338,901,154,638,440 & 846,741,248,149,925,302,187,710 & 846,748,393,995,505,190,521,210 & 846,748,975,531,460,279,032,360 & 846,749,012,589,759,525,290,650 & 846,749,014,437,777,616,142,440 & 846,749,014,509,601,496,535,640 & 846,749,014,511,759,076,621,340 & 846,749,014,511,808,478,701,340 & 846,749,014,511,809,322,004,346 & 846,749,014,511,809,332,363,436 & 846,749,014,511,809,332,449,711 & 846,749,014,511,809,332,450,146 & 846,749,014,511,809,332,450,147 \\\hline \end{array} \]
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 2 & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ \hline 3 & 0 & 1 & 4 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5\\ \hline 4 & 0 & 1 & 8 & 14 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15\\ \hline 5 & 0 & 1 & 16 & 41 & 51 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52 & 52\\ \hline 6 & 0 & 1 & 32 & 122 & 187 & 202 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203 & 203\\ \hline 7 & 0 & 1 & 64 & 365 & 715 & 855 & 876 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877 & 877\\ \hline 8 & 0 & 1 & 128 & 1,094 & 2,795 & 3,845 & 4,111 & 4,139 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140 & 4,140\\ \hline 9 & 0 & 1 & 256 & 3,281 & 11,051 & 18,002 & 20,648 & 21,110 & 21,146 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147 & 21,147\\ \hline 10 & 0 & 1 & 512 & 9,842 & 43,947 & 86,472 & 109,299 & 115,179 & 115,929 & 115,974 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975 & 115,975\\ \hline 11 & 0 & 1 & 1,024 & 29,525 & 175,275 & 422,005 & 601,492 & 665,479 & 677,359 & 678,514 & 678,569 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570 & 678,570\\ \hline 12 & 0 & 1 & 2,048 & 88,574 & 700,075 & 2,079,475 & 3,403,127 & 4,030,523 & 4,189,550 & 4,211,825 & 4,213,530 & 4,213,596 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597 & 4,213,597\\ \hline 13 & 0 & 1 & 4,096 & 265,721 & 2,798,251 & 10,306,752 & 19,628,064 & 25,343,488 & 27,243,100 & 27,602,602 & 27,641,927 & 27,644,358 & 27,644,436 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437 & 27,644,437\\ \hline 14 & 0 & 1 & 8,192 & 797,162 & 11,188,907 & 51,263,942 & 114,700,315 & 164,029,595 & 184,941,915 & 190,077,045 & 190,829,797 & 190,895,863 & 190,899,230 & 190,899,321 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322 & 190,899,322\\ \hline 15 & 0 & 1 & 16,384 & 2,391,485 & 44,747,435 & 255,514,355 & 676,207,628 & 1,084,948,961 & 1,301,576,801 & 1,368,705,291 & 1,381,367,941 & 1,382,847,419 & 1,382,953,889 & 1,382,958,439 & 1,382,958,544 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545 & 1,382,958,545\\ \hline 16 & 0 & 1 & 32,768 & 7,174,454 & 178,973,355 & 1,275,163,905 & 4,010,090,463 & 7,291,973,067 & 9,433,737,120 & 10,254,521,370 & 10,448,276,360 & 10,477,213,268 & 10,479,970,386 & 10,480,136,006 & 10,480,142,026 & 10,480,142,146 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147 & 10,480,142,147\\ \hline 17 & 0 & 1 & 65,536 & 21,523,361 & 715,860,651 & 6,368,612,302 & 23,874,362,200 & 49,582,466,986 & 69,998,462,014 & 79,527,284,317 & 82,285,618,467 & 82,797,679,445 & 82,859,701,769 & 82,864,611,947 & 82,864,861,847 & 82,864,869,667 & 82,864,869,803 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804 & 82,864,869,804\\ \hline 18 & 0 & 1 & 131,072 & 64,570,082 & 2,863,377,067 & 31,821,472,612 & 142,508,723,651 & 339,971,207,051 & 529,007,272,061 & 635,182,667,816 & 672,294,831,619 & 680,685,836,527 & 681,942,165,393 & 682,068,020,031 & 682,076,428,809 & 682,076,796,009 & 682,076,806,005 & 682,076,806,158 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159 & 682,076,806,159\\ \hline 19 & 0 & 1 & 262,144 & 193,710,245 & 11,453,377,195 & 159,042,661,905 & 852,124,263,684 & 2,345,048,898,523 & 4,054,799,902,003 & 5,199,414,528,808 & 5,676,711,562,593 & 5,806,124,780,384 & 5,829,591,731,684 & 5,832,484,170,844 & 5,832,727,748,374 & 5,832,741,665,152 & 5,832,742,192,288 & 5,832,742,204,885 & 5,832,742,205,056 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057 & 5,832,742,205,057\\ \hline 20 & 0 & 1 & 524,288 & 581,130,734 & 45,813,246,635 & 795,019,337,135 & 5,101,098,232,519 & 16,244,652,278,171 & 31,415,584,940,850 & 43,426,867,585,575 & 49,344,452,550,230 & 51,245,294,979,716 & 51,656,311,613,107 & 51,717,380,273,487 & 51,723,682,798,067 & 51,724,135,127,267 & 51,724,157,478,221 & 51,724,158,219,506 & 51,724,158,235,181 & 51,724,158,235,371 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372 & 51,724,158,235,372\\ \hline 21 & 0 & 1 & 1,048,576 & 1,743,392,201 & 183,252,462,251 & 3,974,515,030,652 & 30,560,194,493,456 & 112,871,151,708,404 & 245,382,167,055,488 & 368,654,643,520,692 & 439,841,775,811,967 & 466,668,627,500,968 & 473,501,669,531,146 & 474,706,578,749,477 & 474,855,882,753,977 & 474,868,970,216,557 & 474,869,780,161,021 & 474,869,815,113,820 & 474,869,816,137,255 & 474,869,816,156,540 & 474,869,816,156,750 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751 & 474,869,816,156,751\\ \hline 22 & 0 & 1 & 2,097,152 & 5,230,176,602 & 733,008,800,427 & 19,870,830,712,482 & 183,176,170,057,707 & 785,938,550,025,147 & 1,928,337,630,016,767 & 3,170,300,933,550,687 & 4,005,444,732,928,641 & 4,371,727,233,798,927 & 4,480,550,589,850,064 & 4,503,047,451,718,545 & 4,506,342,616,999,876 & 4,506,688,232,943,076 & 4,506,714,279,517,080 & 4,506,715,683,659,127 & 4,506,715,737,033,756 & 4,506,715,738,423,606 & 4,506,715,738,447,091 & 4,506,715,738,447,322 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323 & 4,506,715,738,447,323\\ \hline 23 & 0 & 1 & 4,194,304 & 15,690,529,805 & 2,932,033,104,555 & 99,348,921,288,655 & 1,098,318,779,272,060 & 5,480,960,778,389,365 & 15,222,915,798,289,765 & 27,542,984,610,086,665 & 37,136,385,907,400,125 & 42,000,637,216,351,225 & 43,672,799,989,835,155 & 44,074,082,550,176,545 & 44,142,711,725,983,660 & 44,151,191,130,412,991 & 44,151,953,491,540,255 & 44,152,003,408,529,058 & 44,152,005,773,414,427 & 44,152,005,853,196,206 & 44,152,005,855,055,756 & 44,152,005,855,084,092 & 44,152,005,855,084,345 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346 & 44,152,005,855,084,346\\ \hline 24 & 0 & 1 & 8,388,608 & 47,071,589,414 & 11,728,128,223,915 & 496,728,911,719,165 & 6,586,964,947,803,695 & 38,264,428,799,608,235 & 120,582,710,957,928,740 & 241,205,285,284,001,240 & 349,459,367,068,932,740 & 412,559,532,764,708,300 & 437,489,737,355,466,560 & 444,378,573,413,388,560 & 445,740,664,435,029,560 & 445,936,484,677,276,640 & 445,957,161,859,742,195 & 445,958,772,809,679,110 & 445,958,865,294,604,555 & 445,958,869,175,343,725 & 445,958,869,292,316,504 & 445,958,869,294,771,110 & 445,958,869,294,805,012 & 445,958,869,294,805,288 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289 & 445,958,869,294,805,289\\ \hline 25 & 0 & 1 & 16,777,216 & 141,214,768,241 & 46,912,504,507,051 & 2,483,597,478,617,802 & 39,510,014,478,620,232 & 267,342,497,477,336,542 & 957,566,218,595,705,122 & 2,125,487,669,688,678,127 & 3,328,651,061,864,065,627 & 4,131,006,966,302,528,287 & 4,493,269,587,087,402,967 & 4,607,754,660,431,147,227 & 4,633,712,770,792,043,227 & 4,638,012,165,447,390,427 & 4,638,538,820,609,086,387 & 4,638,586,883,940,479,497 & 4,638,590,159,619,074,422 & 4,638,590,325,838,044,097 & 4,638,590,332,058,238,847 & 4,638,590,332,226,758,352 & 4,638,590,332,229,958,802 & 4,638,590,332,229,999,052 & 4,638,590,332,229,999,352 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353 & 4,638,590,332,229,999,353\\ \hline 26 & 0 & 1 & 33,554,432 & 423,644,304,722 & 187,650,001,250,987 & 12,417,846,161,543,552 & 237,013,033,135,668,883 & 1,868,866,831,126,685,483 & 7,618,489,083,072,350,433 & 18,820,005,864,027,476,058 & 32,019,561,236,874,324,063 & 42,048,639,577,872,800,823 & 47,198,146,931,729,759,643 & 49,048,715,505,983,309,703 & 49,526,614,124,379,597,963 & 49,617,063,154,570,701,963 & 49,629,789,031,813,184,523 & 49,631,132,763,608,563,353 & 49,631,239,789,154,665,113 & 49,631,246,222,993,683,863 & 49,631,246,513,616,548,538 & 49,631,246,523,375,652,893 & 49,631,246,523,614,582,298 & 49,631,246,523,618,708,498 & 49,631,246,523,618,755,948 & 49,631,246,523,618,756,273 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274 & 49,631,246,523,618,756,274\\ \hline 27 & 0 & 1 & 67,108,864 & 1,270,932,914,165 & 750,599,971,449,515 & 62,088,807,129,858,605 & 1,421,890,125,134,903,156 & 13,069,461,898,046,144,687 & 60,698,293,711,602,480,887 & 167,261,566,992,144,276,462 & 310,458,637,501,567,882,137 & 433,978,054,625,397,974,502 & 505,801,221,212,679,957,102 & 535,008,120,031,833,066,702 & 543,549,269,263,634,652,402 & 545,383,903,334,897,500,662 & 545,677,966,400,968,325,622 & 545,713,535,718,732,248,292 & 545,716,805,910,357,458,802 & 545,717,035,178,844,916,812 & 545,717,047,425,141,229,062 & 545,717,047,920,705,285,192 & 545,717,047,935,720,836,457 & 545,717,047,936,054,668,462 & 545,717,047,936,059,933,462 & 545,717,047,936,059,989,037 & 545,717,047,936,059,989,388 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389 & 545,717,047,936,059,989,389\\ \hline 28 & 0 & 1 & 134,217,728 & 3,812,798,742,494 & 3,002,399,818,689,195 & 310,442,764,649,269,995 & 8,530,588,879,837,946,391 & 91,423,392,608,221,681,659 & 484,101,618,889,583,612,790 & 1,490,799,910,228,016,109,165 & 3,029,333,888,602,793,961,490 & 4,531,244,547,474,348,583,180 & 5,516,641,963,645,562,466,745 & 5,968,154,814,881,834,874,145 & 6,116,937,802,946,210,183,545 & 6,152,998,463,246,954,493,145 & 6,159,538,106,375,350,540,765 & 6,160,436,847,843,408,051,115 & 6,160,531,280,610,425,762,965 & 6,160,538,906,903,312,675,665 & 6,160,539,381,097,726,378,675 & 6,160,539,403,750,867,869,655 & 6,160,539,404,576,774,053,615 & 6,160,539,404,599,467,740,995 & 6,160,539,404,599,927,933,000 & 6,160,539,404,599,934,587,375 & 6,160,539,404,599,934,652,076 & 6,160,539,404,599,934,652,454 & 6,160,539,404,599,934,652,455 & 6,160,539,404,599,934,652,455 & 6,160,539,404,599,934,652,455\\ \hline 29 & 0 & 1 & 268,435,456 & 11,438,396,227,481 & 12,009,599,140,539,051 & 1,552,210,010,313,389,752 & 51,180,527,066,276,028,928 & 639,650,299,280,150,852,200 & 3,863,968,913,259,430,036,516 & 13,316,931,761,586,684,435,022 & 29,708,969,836,672,895,454,647 & 47,768,521,062,634,774,145,562 & 61,095,200,715,560,895,370,032 & 67,950,265,197,803,650,549,797 & 70,484,739,881,941,177,288,797 & 71,174,432,774,516,717,242,197 & 71,315,127,724,871,798,313,717 & 71,336,945,972,957,172,037,287 & 71,339,544,504,231,548,360,937 & 71,339,783,836,563,417,414,087 & 71,339,800,946,744,578,386,987 & 71,339,801,896,654,963,400,577 & 71,339,801,937,478,040,938,677 & 71,339,801,938,825,901,932,377 & 71,339,801,938,859,640,227,877 & 71,339,801,938,860,266,779,257 & 71,339,801,938,860,275,115,858 & 71,339,801,938,860,275,190,765 & 71,339,801,938,860,275,191,171 & 71,339,801,938,860,275,191,172 & 71,339,801,938,860,275,191,172\\ \hline 30 & 0 & 1 & 536,870,912 & 34,315,188,682,442 & 48,038,396,293,720,747 & 7,761,038,612,902,285,822 & 307,071,141,359,850,971,579 & 4,475,987,863,912,937,373,659 & 30,859,006,547,961,045,671,459 & 119,159,990,796,885,614,442,329 & 292,533,334,396,074,979,037,085 & 507,580,435,956,741,855,656,775 & 685,560,143,017,817,189,041,330 & 788,002,660,939,899,127,602,745 & 830,340,371,000,067,257,128,510 & 843,220,239,072,837,883,168,510 & 846,161,051,171,094,720,266,230 & 846,672,656,338,901,154,638,440 & 846,741,248,149,925,302,187,710 & 846,748,393,995,505,190,521,210 & 846,748,975,531,460,279,032,360 & 846,749,012,589,759,525,290,650 & 846,749,014,437,777,616,142,440 & 846,749,014,509,601,496,535,640 & 846,749,014,511,759,076,621,340 & 846,749,014,511,808,478,701,340 & 846,749,014,511,809,322,004,346 & 846,749,014,511,809,332,363,436 & 846,749,014,511,809,332,449,711 & 846,749,014,511,809,332,450,146 & 846,749,014,511,809,332,450,147 \\\hline \end{array} \]
\(n<0\lor k<0\)のとき\(B\left(n,k\right)=0\)の証明
\(n<0\land0\leq k\)のとき、
\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =\sum_{j=0}^{k}0\\ & =0 \end{align*}\(0\leq n\land k<0\)のとき、
\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =-\sum_{j=k+1}^{-1}S_{2}\left(n,j\right)\\ & =0 \end{align*}\(n<0\land k<0\)のとき、
\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =-\sum_{j=k+1}^{-1}S_{2}\left(n,j\right)\\ & =0 \end{align*}-
これより、\begin{align*} \left(n<0\land0\leq k\right)\lor\left(0\leq n\land k<0\right)\lor\left(n<0\land k<0\right) & \Leftrightarrow\left(n<0\land0\leq k\right)\lor k<0\\ & \Leftrightarrow n<0\lor k<0 \end{align*} となるので\(n<0\lor k<0\)のとき、\(B\left(n,k\right)=0\)となる。
ベル数の値
区別の出来ない箱に分けるときの空箱ありと空箱なしとの関係より、空箱なしの場合は\(S_{2}\left(n,j\right)\)なので、\begin{align*} B\left(n,k\right) & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =\sum_{j=0}^{k}S_{2}\left(n,j\right)\\ & =\sum_{j=0}^{k}\frac{1}{j!}\sum_{m=0}^{j}\left(-1\right)^{j-m}C\left(j,m\right)m^{n}\\ & =\sum_{j=0}^{k}\frac{1}{j!}\sum_{m=0}^{j}\left(-1\right)^{j-m}\frac{j!}{m!\left(j-m\right)!}m^{n}\\ & =\sum_{j=0}^{k}\sum_{m=0}^{j}\left(-1\right)^{j-m}\frac{1}{m!\left(j-m\right)!}m^{n}\\ & =\sum_{m=0}^{k}\sum_{j=m}^{k}\left(-1\right)^{j-m}\frac{1}{m!\left(j-m\right)!}m^{n}\\ & =\sum_{m=0}^{k}\sum_{j=0}^{k-m}\left(-1\right)^{j}\frac{1}{m!j!}m^{n}\\ & =\sum_{m=0}^{k}\frac{m^{n}}{m!}\sum_{j=0}^{k-m}\frac{\left(-1\right)^{j}}{j!} \end{align*} となる。
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ベル数の漸化式
\[
B\left(n+1\right)=\sum_{k=0}^{n}C\left(n,k\right)B\left(k\right)
\]
ベル数の指数型母関数
\[
\sum_{k=0}^{\infty}B\left(k\right)\frac{x^{k}}{k!}=e^{e^{x}-1}
\]