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A116218 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks (or pairs) then a(n) is equal to the number of permutations f of X such that f(X_i) != X_i for all i=1,...n.

%I A116218 #23 Aug 06 2024 08:55:57
%S A116218 1,0,20,592,35088,3252608,437765440,80766186240,19580003614976,
%T A116218 6038002429456384,2308538525796209664,1071858241055770480640,
%U A116218 594103565746026102722560,387504996819754568329494528,293818792387460667662661926912,256273357771747968541309427187712
%N A116218 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks (or pairs) then a(n) is equal to the number of permutations f of X such that f(X_i) != X_i for all i=1,...n.
%H A116218 G. C. Greubel, <a href="/A116218/b116218.txt">Table of n, a(n) for n = 0..220</a>
%H A116218 Milan Janjic, <a href="https://web.archive.org/web/20130527052801/http://www.pmfbl.org/janjic/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%F A116218 a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*(2*n-2*i)!.
%F A116218 Recurrence: a(n) = 2*(n-1)*(2*n+1)*a(n-1) + 4*(n-1)*(4*n-3)*a(n-2) + 16*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Mar 20 2014
%F A116218 a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n). - _Vaclav Kotesovec_, Mar 20 2014
%p A116218 a:=n->sum((-2)^i*binomial(n,i)*(2*n-2*i)!,i=0..n);
%t A116218 Table[Sum[(-2)^i*Binomial[n,i]*(2*n-2*i)!,{i,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A116218 (PARI) for(n=1,25, print1(sum(i=0,n, (-2)^i*binomial(n,i)*(2*n-2*i)!), ", ")) \\ _G. C. Greubel_, Mar 18 2017
%Y A116218 Cf. A374980.
%K A116218 nonn
%O A116218 0,3
%A A116218 _Milan Janjic_, Apr 08 2007, corrected Apr 13 2007
%E A116218 a(0)=1 prepended by _Alois P. Heinz_, Aug 05 2024