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

%I A321189 #23 Jan 24 2024 16:20:11
%S A321189 1,1,5,73,2169,108901,8288293,890380177,128364028145,23918924529901,
%T A321189 5595490598128221,1605718043992482553,554663179293965398825,
%U A321189 227038711419826844827381,108674023653792712066606229,60142879347501714200454327841,38108071228342727619600464659425
%N A321189 a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).
%H A321189 Seiichi Manyama, <a href="/A321189/b321189.txt">Table of n, a(n) for n = 0..200</a>
%H A321189 Robert Gill, <a href="https://doi.org/10.1016/S0012-365X(97)00187-8">The number of elements in a generalized partition semilattice</a>, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
%F A321189 a(0) = 1; a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*n^(k-1).
%F A321189 a(n) = A257565(n, n).
%F A321189 From _Vaclav Kotesovec_, Oct 29 2018: (Start)
%F A321189 a(n) ~ exp(1/2) * n! * n^(n-1).
%F A321189 a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n - 1/2). (End)
%F A321189 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
%p A321189 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
%p A321189 # Or, using the recurrence of the Fubini polynomials:
%p A321189 F := proc(n) option remember; if n = 0 then return 1 fi;
%p A321189 expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
%p A321189 a := n -> `if`(n=0, 1, subs(x = n, F(n)) / n):
%p A321189 seq(a(n), n = 0..16);  # Peter Luschny, May 21 2021
%t A321189 Table[n! SeriesCoefficient[1 - 1/(n - 1/(Exp[x] - 1)), {x, 0, n}], {n, 0, 16}]
%t A321189 Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(k - 1), {k, n}], {n, 16}]]
%o A321189 (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
%o A321189 (PARI) {a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 2)))} \\ _Seiichi Manyama_, Jun 12 2020
%Y A321189 Main diagonal of A257565.
%Y A321189 Cf. A000670, A050351, A050352, A050353, A094420.
%K A321189 nonn
%O A321189 0,3
%A A321189 _Ilya Gutkovskiy_, Oct 29 2018