A346055 Expansion of e.g.f. Product_{k>=1} B(x^k/k) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

1, 1, 3, 10, 47, 246, 1617, 11586, 97463, 891610, 9189623, 102024396, 1250714445, 16351489116, 232261545869, 3499469551402, 56582677946675, 964734301550142, 17509882651329087, 333381717125596692, 6710286637806825557, 141167551783524139468

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..449

Formula

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

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.

Prog

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

Crossrefs

Cf. A000110, A005225, A081066, A209903, A346056, A346057.

Sequence in context: A226879 A226880 A005651 * A249479 A355154 A236410

Adjacent sequences: A346052 A346053 A346054 * A346056 A346057 A346058

Keyword

nonn

Author

Seiichi Manyama, Jul 02 2021

Status

approved