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

0,5

Links

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

Formula

T(n, k) = binomial(n, k)*binomial(2*n, 2*k)*f(n)/(f(k)*f[n-k)), where f(n) = Product_{j=1..n+1} j.

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A103371, A155495.

Sequence in context: A144405 A202671 A203004 * A202677 A179838 A174678

Adjacent sequences: A155494 A155495 A155496 * A155498 A155499 A155500

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, May 29 2021

Status

approved