A115179 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

Offset: 0

Author

Paul Barry, Mar 14 2006

Keywords

• easy
• sign
• tabl

Comments

Since C(x*(1-x)) = 1/(1-x), the row sums of this triangle are (1,1,1,...). This establishes the identity Sum_{k=0..n} T(n, k) = Sum_{k=0..n} (-1)^(n-k)*A000108(k)*binomial(k,n-k) = 1.

Examples

			Triangle begins
  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

Crossrefs

Cf. A000012, A000108, A115178, A117434.

Programs

Magma

[(-1)^(n+k)*Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021

Mathematica

Table[(-1)^(n+k)*CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 31 2021 *)

Sage

flatten([[(-1)^(n+k)*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) = (-1)^(n-k)*binomial(k, n-k)*Catalan(k).

Sum_{k=0..n} T(n, k) = A000012(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A115178(n) (upward diagonal sums).

T(n, k) = (-1)^(n+k)*A117434(n, k).