A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

1, 0, 1, 0, 1, 2, 0, 0, 4, 5, 0, 0, 2, 15, 14, 0, 0, 0, 15, 56, 42, 0, 0, 0, 5, 84, 210, 132, 0, 0, 0, 0, 56, 420, 792, 429, 0, 0, 0, 0, 14, 420, 1980, 3003, 1430, 0, 0, 0, 0, 0, 210, 2640, 9009, 11440, 4862, 0, 0, 0, 0, 0, 42, 1980, 15015, 40040, 43758, 16796, 0, 0, 0, 0, 0, 0, 792, 15015, 80080, 175032, 167960, 58786

Offset

0,6

Links

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

Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Formula

T(n, k) = binomial(k, n-k)*Catalan(k).

Sum_{k=0..n} T(n, k) = A052709(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A115178(n) (upward diagonal sums).

T(n, k) = (-1)^(n+k)*A115179(n, k).

Example

Triangle begins as:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 4,  5;
  0, 0, 2, 15, 14;
  0, 0, 0, 15, 56,  42;
  0, 0, 0,  5, 84, 210, 132;
  0, 0, 0,  0, 56, 420, 792, 429;

Mathematica

Table[CatalanNumber[k]*Binomial[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)

Prog

(Magma) [Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021
(Sage) flatten([[binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021

Crossrefs

Cf. A000108, A052709, A115178, A115178.

Sequence in context: A094295 A085969 A115179 * A131742 A257813 A278280

Adjacent sequences: A117431 A117432 A117433 * A117435 A117436 A117437

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 14 2006

Status

approved