0,3

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.

(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)))}

Cf. A125222.

Adjacent sequences: A125220 A125221 A125222 this_sequence A125224 A125225 A125226

Sequence in context: A107593 A125277 A143426 this_sequence A005362 A059439 A006014

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 24 2006