A110511 Riordan array (1/(1+x), x(1-x)/(1+x)^2).

1, -1, 1, 1, -4, 1, -1, 9, -7, 1, 1, -16, 26, -10, 1, -1, 25, -70, 52, -13, 1, 1, -36, 155, -190, 87, -16, 1, -1, 49, -301, 553, -403, 131, -19, 1, 1, -64, 532, -1372, 1462, -736, 184, -22, 1, -1, 81, -876, 3024, -4446, 3206, -1216, 246, -25, 1, 1, -100, 1365, -6084, 11826, -11584, 6190, -1870, 317, -28, 1, -1, 121, -2035

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• sign
• tabl

Comments

Inverse of number triangle A110506. Row sums are A110512. Diagonal sums are A110513. Product of (1/(1+x), x/(1+x)) (inverse binomial transform matrix) and (1, x(1-2x)) (A110509).

Examples

			Rows begin
   1;
  -1,   1;
   1,  -4,   1;
  -1,   9,  -7,   1;
   1, -16,  26, -10,   1;
  -1,  25, -70,  52, -13,   1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Mathematica

T[n_, k_] := Sum[(-1)^(n - j)*Binomial[n, j]*(-2)^(j - k)*Binomial[k, j - k], {j, 0, n}]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

PARI

for(n=0,20, for(k=0,n, print1(sum(j=0,n, (-1)^(n-j)*binomial(n, j)*(-2)^(j-k)*binomial(k, j-k)), ", "))) \\ G. C. Greubel, Aug 29 2017

Formula

Number triangle: T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).

T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k).

T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014