A354505 Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^(1/k!) )^exp(x).

1, 1, 3, 13, 54, 291, 1778, 12167, 82869, 655100, 5658257, 51691806, 454932679, 4527660281, 48270581011, 553646849053, 5561424579562, 72988254250439, 1010390962699396, 12295679951427509, 67360732923382327, 1515500302797716376, 45199587363022824107, 1001538050395504921200, -699211952404047871075

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354509(k) * binomial(n-1,k-1) * a(n-k).

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k!))^exp(x)))
(PARI) a354509(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!))/(n-k)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354509(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Crossrefs

Cf. A354509, A356402.

Sequence in context: A323264 A362495 A305655 * A308349 A367153 A140320

Adjacent sequences: A354502 A354503 A354504 * A354506 A354507 A354508

Keyword

sign

Author

Seiichi Manyama, Aug 15 2022

Status

approved