Pyramid #716
Generating function
$$U_{716}(x, y) = \sqrt{\frac{4 x + 1}{1 - 4 y}}$$
Explicit formula
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{716}(n, m, k) = \operatorname{TA_{984}}{\left(m,k \right)} \operatorname{Tsqrt}{\left(n,k \right)}$$
Data table
1 2 6 20 70 252 924
2 4 12 40 140 504 1848
-2 -4 -12 -40 -140 -504 -1848
4 8 24 80 280 1008 3696
-10 -20 -60 -200 -700 -2520 -9240
28 56 168 560 1960 7056 25872
-84 -168 -504 -1680 -5880 -21168 -77616
Related
Export
expand_less