A155516 Triangle T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1), read by rows.

1, 1, 1, 1, 20, 1, 1, 105, 105, 1, 1, 336, 1764, 336, 1, 1, 825, 13860, 13860, 825, 1, 1, 1716, 70785, 226512, 70785, 1716, 1, 1, 3185, 273273, 2147145, 2147145, 273273, 3185, 1, 1, 5440, 866320, 14158144, 34763300, 14158144, 866320, 5440, 1, 1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1

Offset

0,5

Links

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

Formula

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

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

T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*A155495(n, k).

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

Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n,-n,-n-1/2,-n+1/2], [1/2,1,3/2], 1). (End)

Example

Triangle begins as:
  1;
  1,    1;
  1,   20,       1;
  1,  105,     105,        1;
  1,  336,    1764,      336,         1;
  1,  825,   13860,    13860,       825,         1;
  1, 1716,   70785,   226512,     70785,      1716,        1;
  1, 3185,  273273,  2147145,   2147145,    273273,     3185,       1;
  1, 5440,  866320, 14158144,  34763300,  14158144,   866320,    5440,    1;
  1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1;

Mathematica

T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[2*n+1, 2*k+1]/(2*n-2*k+1);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 29 2021 *)

Prog

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

Crossrefs

Cf. A155495.

Sequence in context: A040400 A139459 A154652 * A174674 A144443 A257608

Adjacent sequences: A155513 A155514 A155515 * A155517 A155518 A155519

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, May 29 2021

Status

approved