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

Author

Paul Barry, Mar 14 2006

Keywords

• easy
• nonn
• tabl

Examples

			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;
		

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.

Crossrefs

Cf. A000108, A052709, A115178, A115178.

Programs

Magma

[Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021

Mathematica

Table[CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* 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

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).