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

Author

Seiichi Manyama, Jul 02 2021

Keywords

• nonn

Links

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

Crossrefs

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

Programs

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)!));

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.