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.

1, 0, 20, 592, 35088, 3252608, 437765440, 80766186240, 19580003614976, 6038002429456384, 2308538525796209664, 1071858241055770480640, 594103565746026102722560, 387504996819754568329494528, 293818792387460667662661926912, 256273357771747968541309427187712

Offset: 0

Author

Milan Janjic, Apr 08 2007, corrected Apr 13 2007

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..220
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.

Crossrefs

Cf. A374980.

Programs

Maple

a:=n->sum((-2)^i*binomial(n,i)*(2*n-2*i)!,i=0..n);

Mathematica

Table[Sum[(-2)^i*Binomial[n,i]*(2*n-2*i)!,{i,0,n}],{n,1,20}] (* Vaclav Kotesovec, Mar 20 2014 *)

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

Formula

a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*(2*n-2*i)!.

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

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Mar 20 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 05 2024