A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385

Offset: 0

Author

Seiichi Manyama, Jun 26 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451

Crossrefs

Cf. A028343, A087906, A185895, A209902, A330199, A345758, A346039, A346057.

Programs

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/k))))

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d-1)!)*x^k/k))))

PARI

a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));

Formula

E.g.f.: exp( -Sum_{k>=1} (exp(x^k) - 1)/k ).

E.g.f.: exp( -Sum_{k>=1} A087906(k)*x^k/k! ).

a(n) = -(n-1)! * Sum_{k=1..n} (Sum_{d|k} 1/(d-1)!) * a(n-k)/(n-k)! for n > 0.