cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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).

Original entry on oeis.org

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

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Author

Peter Luschny, Nov 07 2023

Keywords

Examples

			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;
		

Crossrefs

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

Programs

  • 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 *)

Formula

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