A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 6, 8, 2, 1, 2, 8, 20, 16, 2, 1, 2, 10, 38, 70, 32, 2, 1, 2, 12, 62, 196, 252, 64, 2, 1, 2, 14, 92, 426, 1062, 924, 128, 2, 1, 2, 16, 128, 792, 3112, 5948, 3432, 256, 2, 1, 2, 18, 170, 1326, 7302, 23686, 34120, 12870, 512, 2

Offset: 0

Author

Paul Barry, May 09 2005

Keywords

• easy
• nonn
• tabl

Comments

As a number square read by antidiagonals, the rows represent the row sums of the inverses of the Riordan arrays (1/(1+x),x/(1+x)^k), k>=0. The rows are then given by T(n,k)=(n-1)C(n*k,k)-(n-2)*sum{j=0..k, C(n*k,j)}.

Examples

			Triangle begins
  1;
  1, 2;
  1, 2, 2;
  1, 2, 4,  2;
  1, 2, 6,  8,  2;
  1, 2, 8, 20, 16, 2;
		

Crossrefs

T(n,n) is A040000, T(n+1,n) is A000079, T(n+2,n) is A000984, T(n+3,n) is A047098.

The reverse of this triangle is A107030.

Row sums are A107028.

Diagonal sums are A107029.

Formula

Number triangle T(n, k)=if(k<=n, (n-k-1)C((n-k)*k, k)-(n-k-2)*sum{j=0..k, C((n-k)*k, j)}, 0).