A346037 Expansion of e.g.f. Product_{k>=1} B(x^k)^(1/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

1, 1, 3, 9, 41, 183, 1145, 6835, 52043, 398441, 3577291, 32395905, 342875813, 3603992759, 42817702673, 518311440987, 6897155535843, 93092680608025, 1376879589495555, 20561329595474713, 333009853668160757, 5480574201430489831, 96322698607644959065

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..464

Formula

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

E.g.f.: exp( Sum_{k>=1} A121860(k)*x^k/k! ).

a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/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)-1)^(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!*(k/d)!))*x^k))))
(PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!))*a(n-k)/(n-k)!));

Crossrefs

Cf. A000110, A121860, A209902, A209903, A346039, A346056.

Sequence in context: A027893 A018417 A228478 * A320918 A325289 A274739

Adjacent sequences: A346034 A346035 A346036 * A346038 A346039 A346040

Keyword

nonn

Author

Seiichi Manyama, Jul 02 2021

Status

approved