A367025 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = (1 - hypergeom([-1/2, -n - 1, -n - 1], [1, 1], 4*x)) / (2*x).

1, 4, 1, 9, 9, 2, 16, 36, 32, 5, 25, 100, 200, 125, 14, 36, 225, 800, 1125, 504, 42, 49, 441, 2450, 6125, 6174, 2058, 132, 64, 784, 6272, 24500, 43904, 32928, 8448, 429, 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

T(n,k) = binomial(n+1,n-k)^2*binomial(2*k,k)/(k+1). - Detlef Meya, Nov 19 2023

Example

Triangle T(n, k) starts:
  [0]   1;
  [1]   4,    1;
  [2]   9,    9,     2;
  [3]  16,   36,    32,      5;
  [4]  25,  100,   200,    125,     14;
  [5]  36,  225,   800,   1125,    504,      42;
  [6]  49,  441,  2450,   6125,   6174,    2058,     132;
  [7]  64,  784,  6272,  24500,  43904,   32928,    8448,    429;
  [8]  81, 1296, 14112,  79380, 222264,  296352,  171072,  34749,   1430;
  [9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;

Maple

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

Mathematica

T[n_, k_]:=Binomial[n+1, n-k]^2*Binomial[2*k, k]/(k+1); Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, Nov 19 2023 *)

Crossrefs

Cf. A000290 (first column), A000108 (main diagonal).

Cf. A367022, A367023, A387024.

Sequence in context: A021990 A182545 A084887 * A067015 A179193 A158199

Adjacent sequences: A367022 A367023 A367024 * A367026 A367027 A367028

Keyword

nonn,tabl

Author

Peter Luschny, Nov 07 2023

Status

approved