A321189 a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).

1, 1, 5, 73, 2169, 108901, 8288293, 890380177, 128364028145, 23918924529901, 5595490598128221, 1605718043992482553, 554663179293965398825, 227038711419826844827381, 108674023653792712066606229, 60142879347501714200454327841, 38108071228342727619600464659425

Offset: 0

Author

Ilya Gutkovskiy, Oct 29 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200
• Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.

Crossrefs

Main diagonal of A257565.

Cf. A000670, A050351, A050352, A050353, A094420.

Programs

GAP

Concatenation([1],List([1..16],n->Sum([1..n],k->Stirling2(n,k)*Factorial(k)*n^(k-1)))); # Muniru A Asiru, Oct 29 2018

Maple

seq(coeff(series(factorial(n)*(1-1/(n-1/(exp(x)-1))),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 29 2018
# Or, using the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> `if`(n=0, 1, subs(x = n, F(n)) / n):
seq(a(n), n = 0..16);  # Peter Luschny, May 21 2021

Mathematica

Table[n! SeriesCoefficient[1 - 1/(n - 1/(Exp[x] - 1)), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(k - 1), {k, n}], {n, 16}]]

PARI

{a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 2)))} \\ Seiichi Manyama, Jun 12 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*n^(k-1).

a(n) = A257565(n, n).

From Vaclav Kotesovec, Oct 29 2018: (Start)

a(n) ~ exp(1/2) * n! * n^(n-1).

a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n - 1/2). (End)

a(n) = F_{n}(n) / n for n >= 1, where F_{n}(x) is the Fubini polynomial. In other words: a(n) = A094420(n) / n for n >= 1. - Peter Luschny, May 21 2021