A125223 - OEIS

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

%S 1,1,2,7,32,175,1091,7540,56744,459379,3965669,36266437,349564610,

%T 3536884843,37440399437,413499135061,4753206442286,56751068291533,

%U 702488064111341,9000518743165651,119183035725216482

%N a(n) = Sum_{k=0..n-1} C(n-1,k)* [x^(n-k-1)] A(x)^(k+1) for n>0, with a(0)=1.

%e A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1091*x^6 +...

%e From the table of n-th self-convolutions:

%e A^1: [1, 1, 2, 7, 32, 175, 1091, 7540, 56744, 459379, ...];

%e A^2: [1, 2, 5, 18, 82, 442, 2709, 18410, 136406, 1088880, ...];

%e A^3: [1, 3, 9, 34, 156, 834, 5042, 33750, 246381, 1939768, ...];

%e A^4: [1, 4, 14, 56, 261, 1392, 8330, 55028, 396178, 3077496, ...];

%e A^5: [1, 5, 20, 85, 405, 2166, 12875, 84115, 597940, 4585270, ...];

%e A^6: [1, 6, 27, 122, 597, 3216, 19052, 123372, 867066, 6568386, ...];

%e illustrate a(n) = Sum_{k=0..n-1} C(n-1,k)*[x^(n-k-1)] A(x)^(k+1) by:

%e a(2) = 1*(1) + 1*(1) = 2;

%e a(3) = 1*(2) + 2*(2) + 1*(1) = 7;

%e a(4) = 1*(7) + 3*(5) + 3*(3) + 1*(1) = 32;

%e a(5) = 1*(32) + 4*(18) + 6*(9) + 4*(4) + 1*(1) = 175.

%o (PARI) {a(n)=local(A=1+sum(k=1,n-1,a(k)*x^k));if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*polcoeff(A^(n-k),k)))}

%Y Cf. A125222.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 24 2006