A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 26, 42, 16, 1, 1, 6, 57, 184, 163, 32, 1, 1, 7, 120, 731, 1358, 638, 64, 1, 1, 8, 247, 2736, 10121, 10244, 2510, 128, 1, 1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1, 1, 10, 1013, 34448, 468735, 1911956, 2141835

Offset: 0

Author

Paul D. Hanna, Nov 09 2006

Keywords

• nonn
• tabl

Examples

			Column g.f.s begin:
C_0(x) = 1/(1-x);
C_1(x) = 1/((1-x)(1-x));
C_2(x) = 1/((1-x)(1-2x)(1-x));
C_3(x) = 1/((1-x)(1-3x)(1-3x)(1-x));
C_4(x) = 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x)); ...
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 11, 8, 1;
1, 5, 26, 42, 16, 1;
1, 6, 57, 184, 163, 32, 1;
1, 7, 120, 731, 1358, 638, 64, 1;
1, 8, 247, 2736, 10121, 10244, 2510, 128, 1;
1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1;
1, 10, 1013, 34448, 468735, 1911956, 2141835, 604160, 39203, 512, 1; ...
		

Crossrefs

Cf. A124835 (row sums), A124836 (central terms).

Programs

PARI

{T(n,k)=polcoeff(1/prod(j=0,k,1-binomial(k,j)*x +x*O(x^n)),n-k)}

Formula

T(n+1,n) = 2^n. T(n+2,n) = A032443(n) = Sum_{i=0..n} binomial(2*n,i).