A352005 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)!).

1, 0, 1, 1, -3, 11, -5, -83, -2919, 18838, 118371, 583826, -27365327, -12780260, 405396069, 32646641041, -232690739007, 4816360930145, -46984166770283, -541620811734953, -49355727191815599, 907100235094018036, 10877428540752188625, 139350853273096742762

Offset: 0

Author

Seiichi Manyama, Feb 28 2022

Keywords

• sign

Crossrefs

Cf. A298906, A352003, A352004, A352014.

Programs

PARI

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

PARI

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

Formula

E.g.f.: exp( Sum_{k>=1} A352014(k)*x^k/k! ) where A352014(k) = Sum_{p|k, p prime} (-1)^(k/p+1) * (k-1)!/(p-1)!.