= Solution
Use the <Cox-de Boor recurrence>. For unit knots the order-two hat is $L(u)=u$ on $[0,1]$, $L(u)=2-u$ on $[1,2]$, and zero elsewhere. The order-three <quadratic cardinal B-spline> is
$$
q(u)=\frac u2L(u)+\frac{3-u}{2}L(u-1)
=\begin{cases}
\tfrac12u^2,&0\le u\le1,\\
\tfrac34-(u-\tfrac32)^2,&1\le u\le2,\\
\tfrac12(3-u)^2,&2\le u\le3,\\
0,&\text{otherwise}.
\end{cases}
$$
Thus $N_j(t)=q(t-j)$. At $x_i=i+3/2$, the only possible arguments in its <support> are $3/2$ and $1/2,5/2$ at neighboring indices. Their values give
$$
\boxed{N_j(x_i)=\begin{cases}\tfrac34,&j=i,\\\tfrac18,&|j-i|=1,\\0,&|j-i|\ge2.\end{cases}}
$$
Indices outside $1,\ldots,n$ are absent, so the first and last rows have just one neighboring nonzero value.
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