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
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; ...
Programs
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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).