cp's OEIS Frontend

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

%I A108459 #37 Dec 10 2024 08:58:01
%S A108459 1,1,5,52,855,19921,614866,24040451,1152972925,66200911138,
%T A108459 4465023867757,348383154017581,31052765897026352,3128792250765898965,
%U A108459 353179564583216567917,44320731930172534543092,6141797839043095806714667,934330605640859569909566925
%N A108459 Number of labeled partitions of (n,n) into pairs (i,j).
%C A108459 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 A108459 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 A108459 Seiichi Manyama, <a href="/A108459/b108459.txt">Table of n, a(n) for n = 0..200</a>
%F A108459 a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - _Vladeta Jovovic_, Aug 31 2006
%F A108459 E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - _Vladeta Jovovic_, Jul 12 2007
%F A108459 E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - _Paul D. Hanna_, Jun 28 2019
%F A108459 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 A108459 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 A108459 b:= proc(n) option remember; expand(`if`(n=0, 1,
%p A108459       x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
%p A108459     end:
%p A108459 a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
%p A108459 seq(a(n), n=0..21);  # _Alois P. Heinz_, Dec 02 2023
%t A108459 a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
%t A108459 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Dec 10 2024 *)
%o A108459 (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 A108459 (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 A108459 Main diagonal of A108458. Cf. A108461.
%Y A108459 Cf. A048993 (Stirling2), A068424 (falling factorial).
%Y A108459 Cf. A326600, A326270, A326271, A326288.
%Y A108459 Bisection of A124421 (even part).
%K A108459 nonn
%O A108459 0,3
%A A108459 _Christian G. Bower_, Jun 03 2005