A125223 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.

1, 1, 2, 7, 32, 175, 1091, 7540, 56744, 459379, 3965669, 36266437, 349564610, 3536884843, 37440399437, 413499135061, 4753206442286, 56751068291533, 702488064111341, 9000518743165651, 119183035725216482

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Example

A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1091*x^6 +...
From the table of n-th self-convolutions:
A^1: [1, 1, 2, 7, 32, 175, 1091, 7540, 56744, 459379, ...];
A^2: [1, 2, 5, 18, 82, 442, 2709, 18410, 136406, 1088880, ...];
A^3: [1, 3, 9, 34, 156, 834, 5042, 33750, 246381, 1939768, ...];
A^4: [1, 4, 14, 56, 261, 1392, 8330, 55028, 396178, 3077496, ...];
A^5: [1, 5, 20, 85, 405, 2166, 12875, 84115, 597940, 4585270, ...];
A^6: [1, 6, 27, 122, 597, 3216, 19052, 123372, 867066, 6568386, ...];
illustrate a(n) = Sum_{k=0..n-1} C(n-1,k)*[x^(n-k-1)] A(x)^(k+1) by:
a(2) = 1*(1) + 1*(1) = 2;
a(3) = 1*(2) + 2*(2) + 1*(1) = 7;
a(4) = 1*(7) + 3*(5) + 3*(3) + 1*(1) = 32;
a(5) = 1*(32) + 4*(18) + 6*(9) + 4*(4) + 1*(1) = 175.

Prog

(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)))}

Crossrefs

Cf. A125222.

Sequence in context: A161392 A161393 A143426 * A339226 A277359 A005362

Adjacent sequences: A125220 A125221 A125222 * A125224 A125225 A125226

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2006

Status

approved