A294464 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(1/k).

1, 1, 2, 12, 36, 300, 2520, 20160, 75600, 2192400, 30996000, 276091200, 2165486400, 19070251200, 968042275200, 41954552640000, 190974944160000, 230641066656000, 95607669148992000, -2052972258809472000, 22839078791168640000, 5074390517301705600000

Offset: 0

Author

Seiichi Manyama, Oct 31 2017

Keywords

• sign

Crossrefs

Cf. A022629, A076717, A168243, A294462.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+k*x^k)^(1/k))))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025