A346056 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, 9, 38, 168, 915, 5225, 34228, 236622, 1805297, 14498751, 125907798, 1146476984, 11129874215, 112934907867, 1209762361679, 13499714095281, 157931096158594, 1918777335806274, 24309294470496502, 318987321135326838, 4346474397776153974

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..520

Formula

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

E.g.f.: exp( Sum_{k>=1} A038041(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, A038041, A081066, A209903, A346055, A346058.

Sequence in context: A149024 A149025 A351070 * A133222 A103942 A030928

Adjacent sequences: A346053 A346054 A346055 * A346057 A346058 A346059

Keyword

nonn

Author

Seiichi Manyama, Jul 02 2021

Status

approved