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A321189
a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).
5
1, 1, 5, 73, 2169, 108901, 8288293, 890380177, 128364028145, 23918924529901, 5595490598128221, 1605718043992482553, 554663179293965398825, 227038711419826844827381, 108674023653792712066606229, 60142879347501714200454327841, 38108071228342727619600464659425
OFFSET
0,3
LINKS
Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
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
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}]]
PROG
(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
(PARI) {a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 2)))} \\ Seiichi Manyama, Jun 12 2020
CROSSREFS
Main diagonal of A257565.
Sequence in context: A048144 A144682 A293146 * A301387 A096987 A096538
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 29 2018
STATUS
approved