(1)

\begin{align*} B\left(0,k\right) & =\sum_{j=0}^{k}S_{2}\left(0,j\right)\\ & =\sum_{j=0}^{k}\delta_{0,j}\\ & =\begin{cases} \sum_{j=0}^{k}\delta_{0,j} & 0\leq k\\ -\sum_{j=k+1}^{-1}\delta_{0,j} & k<0 \end{cases}\\ & =\begin{cases} 1 & 0\leq k\\ 0 & k<0 \end{cases}\\ & =H\left(\frac{1}{2}+k\right) \end{align*}

(2)

\begin{align*} B\left(1,k\right) & =\sum_{j=0}^{k}S_{2}\left(1,j\right)\\ & =\sum_{j=0}^{k}\delta_{1,j}\\ & =\begin{cases} \sum_{j=0}^{k}\delta_{1,j} & 0\leq k\\ -\sum_{j=k+1}^{-1}\delta_{1,j} & k<0 \end{cases}\\ & =\begin{cases} H\left(-\frac{1}{2}+k\right) & 0\leq k\\ 0 & k<0 \end{cases}\\ & =H\left(-\frac{1}{2}+k\right) \end{align*}

(3)

\begin{align*} B\left(n,0\right) & =\sum_{j=0}^{0}S_{2}\left(n,j\right)\\ & =S_{2}\left(n,0\right)\\ & =\delta_{0,n} \end{align*}

(4)

\begin{align*} B\left(n,1\right) & =\sum_{j=0}^{1}S_{2}\left(n,j\right)\\ & =S_{2}\left(n,0\right)+S_{2}\left(n,1\right)\\ & =\delta_{0,n}+H\left(-\frac{1}{2}+n\right)\\ & =H\left(\frac{1}{2}+n\right) \end{align*}

(5)

\begin{align*} B\left(n,2\right) & =\sum_{j=0}^{2}S_{2}\left(n,j\right)\\ & =S_{2}\left(n,0\right)+S_{2}\left(n,1\right)+S_{2}\left(n,2\right)\\ & =\delta_{0,n}+H\left(-\frac{1}{2}+n\right)+\left(2^{n-1}-1\right)H\left(-\frac{1}{2}+n\right)\\ & =\delta_{0,n}+2^{n-1}H\left(-\frac{1}{2}+n\right) \end{align*}

(6)

\begin{align*} B\left(n,n\right) & =\sum_{j=0}^{n}S_{2}\left(n,j\right)\\ & =\sum_{j=0}^{n-1}S_{2}\left(n,j\right)+S_{2}\left(n,n\right)\\ & =B\left(n,n-1\right)+H\left(\frac{1}{2}+n\right) \end{align*}