A124834 - OEIS

%I #3 Mar 30 2012 18:37:01

%S 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,

%T 7,120,731,1358,638,64,1,1,8,247,2736,10121,10244,2510,128,1,1,9,502,

%U 9844,70436,145475,78320,9908,256,1,1,10,1013,34448,468735,1911956,2141835

%N 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).

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

%e Column g.f.s begin:

%e C_0(x) = 1/(1-x);

%e C_1(x) = 1/((1-x)(1-x));

%e C_2(x) = 1/((1-x)(1-2x)(1-x));

%e C_3(x) = 1/((1-x)(1-3x)(1-3x)(1-x));

%e C_4(x) = 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x)); ...

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 4, 1;

%e 1, 4, 11, 8, 1;

%e 1, 5, 26, 42, 16, 1;

%e 1, 6, 57, 184, 163, 32, 1;

%e 1, 7, 120, 731, 1358, 638, 64, 1;

%e 1, 8, 247, 2736, 10121, 10244, 2510, 128, 1;

%e 1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1;

%e 1, 10, 1013, 34448, 468735, 1911956, 2141835, 604160, 39203, 512, 1; ...

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

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 09 2006