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%I #33 Dec 04 2023 15:04:57
%S 1,1,5,52,855,19921,614866,24040451,1152972925,66200911138,
%T 4465023867757,348383154017581,31052765897026352,3128792250765898965,
%U 353179564583216567917,44320731930172534543092,6141797839043095806714667,934330605640859569909566925
%N Number of labeled partitions of (n,n) into pairs (i,j).
%C Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.
%C a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - _James East_, Apr 10 2018
%H Seiichi Manyama, <a href="/A108459/b108459.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - _Vladeta Jovovic_, Aug 31 2006
%F E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - _Vladeta Jovovic_, Jul 12 2007
%F E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - _Paul D. Hanna_, Jun 28 2019
%F O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - _Paul D. Hanna_, Sep 17 2013
%F a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - _James East_, Apr 10 2018
%p b:= proc(n) option remember; expand(`if`(n=0, 1,
%p x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
%p end:
%p a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
%p seq(a(n), n=0..21); # _Alois P. Heinz_, Dec 02 2023
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ _Paul D. Hanna_, Sep 17 2013
%o (PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ _Paul D. Hanna_, Sep 17 2013
%Y Main diagonal of A108458. Cf. A108461.
%Y Cf. A048993 (Stirling2), A068424 (falling factorial).
%Y Cf. A326600, A326270, A326271, A326288.
%Y Bisection of A124421 (even part).
%K nonn
%O 0,3
%A _Christian G. Bower_, Jun 03 2005
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