A154948 Riordan array ((1+x)/(1-x^2)^2, x(1+x)/(1-x)).

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 14, 7, 1, 3, 15, 30, 26, 9, 1, 4, 21, 55, 70, 42, 11, 1, 4, 28, 91, 155, 138, 62, 13, 1, 5, 36, 140, 301, 363, 242, 86, 15, 1, 5, 45, 204, 532, 819, 743, 390, 114, 17, 1, 6, 55, 285, 876, 1652, 1925, 1375, 590, 146, 19, 1

Offset: 0

Author

Paul Barry, Jan 17 2009

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A113300(n+1). Diagonal sums are A154949.

Product of A154950 and A007318.

Examples

			Triangle begins
  1;
  1,  1;
  2,  3,  1;
  2,  6,  5,  1;
  3, 10, 14,  7,  1;
  3, 15, 30, 26,  9,  1;
  4, 21, 55, 70, 42, 11, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

Magma

[ (&+[Binomial(k-1, j)*Binomial(n-j+1, k+1): j in [0..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

Maple

seq(seq( add(binomial(k-1, j)*binomial(n-j+1, k+1), j=0..n+1), k=0..n), n=0..10); # G. C. Greubel, Feb 18 2020

Mathematica

Table[Binomial[n+1, k+1]*Hypergeometric2F1[-n+k, -k+1, -n-1, -1], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

Sage

[[ sum(binomial(k-1, j)*binomial(n-j+1, k+1) for j in (0..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020

Formula

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

T(n, k) = binomial(n+1,k+1)*2F1(-(n-k), -(k-1); -(n+1); -1). - G. C. Greubel, Feb 18 2020

Extensions

a(45)=0 removed by Georg Fischer, Feb 18 2020