A346893 Expansion of e.g.f. 1 / (1 - x^5 * exp(x) / 5!).

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 504, 6006, 67320, 577863, 4038034, 24975951, 165481680, 1553590220, 19495772856, 249507077436, 2910465717648, 31103684847837, 326286335505438, 3766644374319673, 51399738264984648, 785038533451101930

Offset: 0

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..498

Crossrefs

Column k=5 of A351703.

Cf. A006153, A346888, A346889, A346890.

Cf. A000389, A145455.

Programs

Mathematica

nmax = 25; CoefficientList[Series[1/(1 - x^5 Exp[x]/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^5*exp(x)/5!))) \\ Michel Marcus, Aug 06 2021

PARI

a(n) = n!*sum(k=0, n\5, k^(n-5*k)/(120^k*(n-5*k)!)); \\ Seiichi Manyama, May 13 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,5) * a(n-k).

a(n) ~ n! / ((1 + LambertW(2^(3/5)*3^(1/5)/5^(4/5))) * 5^(n+1) * LambertW(2^(3/5)*3^(1/5)/5^(4/5))^n). - Vaclav Kotesovec, Aug 08 2021

a(n) = n! * Sum_{k=0..floor(n/5)} k^(n-5*k)/(120^k * (n-5*k)!). - Seiichi Manyama, May 13 2022