A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).

1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870

Offset: 0

Author

Peter Luschny, Nov 07 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,   2;
  [2] 1,   8,    6;
  [3] 1,  18,   54,     20;
  [4] 1,  32,  216,    320,      70;
  [5] 1,  50,  600,   2000,    1750,     252;
  [6] 1,  72, 1350,   8000,   15750,    9072,     924;
  [7] 1,  98, 2646,  24500,   85750,  111132,   45276,    3432;
  [8] 1, 128, 4704,  62720,  343000,  790272,  724416,  219648,   12870;
  [9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
		

Crossrefs

Cf. A002893 (row sum), A002897 (central column), A000984 (main diagonal).

Cf. A367022, A367023, A387024.

Programs

Maple

p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) = binomial(n, k)^2 * binomial(2*k, k).