A346057 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).

1, -1, -1, 2, 3, 14, -55, 62, -637, 338, -3861, 335312, -4499803, 43490108, -246353731, 2189950310, -47336985225, 1224524919590, -21516426400621, 346681988108648, -4499477383730851, 69294602646065900, -1418045089870455795, 45246859024830444566

Offset: 0

Author

Seiichi Manyama, Jul 02 2021

Keywords

• sign

Links

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

Crossrefs

Cf. A000587, A005225, A330199, A346055, A346058.

Programs

PARI

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

PARI

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

PARI

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

PARI

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

Formula

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

E.g.f.: exp( -Sum_{k>=1} A005225(k)*x^k/k! ).

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