A155497 Triangle T(n, k) = binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1), read by rows.

1, 1, 1, 1, 18, 1, 1, 90, 90, 1, 1, 280, 1400, 280, 1, 1, 675, 10500, 10500, 675, 1, 1, 1386, 51975, 161700, 51975, 1386, 1, 1, 2548, 196196, 1471470, 1471470, 196196, 2548, 1, 1, 4320, 611520, 9417408, 22702680, 9417408, 611520, 4320, 1, 1, 6885, 1652400, 46781280, 231567336, 231567336, 46781280, 1652400, 6885, 1

Offset: 0

Author

Roger L. Bagula, Jan 23 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   18,       1;
  1,   90,      90,        1;
  1,  280,    1400,      280,         1;
  1,  675,   10500,    10500,       675,         1;
  1, 1386,   51975,   161700,     51975,      1386,        1;
  1, 2548,  196196,  1471470,   1471470,    196196,     2548,       1;
  1, 4320,  611520,  9417408,  22702680,   9417408,   611520,    4320,    1;
  1, 6885, 1652400, 46781280, 231567336, 231567336, 46781280, 1652400, 6885, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A103371, A155495.

Programs

Magma

[Binomial(n, k)*Binomial(n+1, k+1)*Binomial(2*n, 2*k)/(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

Mathematica

T[n_, k_]:= Binomial[n,k]*Binomial[n+1,k+1]*Binomial[2*n,2*k]/(n-k+1);
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 29 2021 *)

Sage

flatten([[binomial(n, k)*binomial(n+1, k+1)*binomial(2*n, 2*k)/(n-k+1) for k in (0..12)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Formula

T(n, k) = binomial(n, k)*binomial(2*n, 2*k)*f(n)/(f(k)*f(n-k)), where f(n) = (n+1)!.

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

From G. C. Greubel, May 29 2021: (Start)

Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n, -n, -n-1, -n-1/2], [1/2, 1, 2], 1).

T(n, k) = binomial(n, k)*A155495(n, k)/(n-k+1).

T(n, k) = binomial(2*n, 2*k)*A103371(n, k)/(n-k+1). (End)

Extensions

Edited by G. C. Greubel, May 29 2021