A367024 Triangle read by rows, T(n, k) = [x^k] -hypergeom([-1/2, -n, -n], [1, 1], 4*x).

-1, -1, 2, -1, 8, 2, -1, 18, 18, 4, -1, 32, 72, 64, 10, -1, 50, 200, 400, 250, 28, -1, 72, 450, 1600, 2250, 1008, 84, -1, 98, 882, 4900, 12250, 12348, 4116, 264, -1, 128, 1568, 12544, 49000, 87808, 65856, 16896, 858, -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (2*k - 1).

Example

Triangle T(n, k) starts:
  [0] -1;
  [1] -1,   2;
  [2] -1,   8,    2;
  [3] -1,  18,   18,     4;
  [4] -1,  32,   72,    64,     10;
  [5] -1,  50,  200,   400,    250,     28;
  [6] -1,  72,  450,  1600,   2250,   1008,     84;
  [7] -1,  98,  882,  4900,  12250,  12348,   4116,    264;
  [8] -1, 128, 1568, 12544,  49000,  87808,  65856,  16896,   858;
  [9] -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860;

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);

Crossrefs

Cf. A246065 (row sums), -A002420 and A284016 (main diagonal).

Cf. A367022, A367023, A387025.

Sequence in context: A180956 A208921 A208660 * A114706 A046740 A317932

Adjacent sequences: A367021 A367022 A367023 * A367025 A367026 A367027

Keyword

sign,tabl

Author

Peter Luschny, Nov 07 2023

Status

approved