A209289 Number of functions f:{1,2,...,2n}->{1,2,...,2n} such that every preimage has an even cardinality.

1, 2, 40, 2256, 250496, 46063360, 12665422848, 4866544707584, 2490379333697536, 1637285952230719488, 1344814260872574402560, 1349528279475362368847872, 1624638302165034485761966080, 2310920106523435237448955723776, 3834278385523271302103123693142016

Offset

0,2

Comments

Note that the empty set has even cardinality.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..210 (terms 0..80 from Alois P. Heinz)

Formula

a(n) = (2n)! * [x^(2n)] cosh(x)^(2n).

a(n) = Sum_{i=0..2*n} (n-i)^(2*n)*binomial(2*n,i). - Vladimir Kruchinin, Feb 07 2013

a(n) ~ c * n^(2*n) * 2^(2*n) * (1-r)^(2*n) / ((2-r)^n * r^n * exp(2*n)), where r = 0.1664434403990353015638385297757806508596082... is the root of the equation (2/r-1)^(1-r) = exp(2), and c = 1.66711311920192939687232294044843869828... = 2/A085984. - Vaclav Kotesovec, Sep 03 2014, updated Mar 18 2024

Example

a(1) = 2 because there are 2 functions from {1,2} into {1,2} for which the preimage of both elements has even size: 1,1 (where the preimage of 1 is {1,2} and the preimage of 2 is the empty set) and 2,2 (where the preimage of 1 is the empty set and the preimage of 2 is {1,2}).

Maple

a:= n-> (2*n)! *coeff(series(cosh(x)^(2*n), x, 2*n+1), x, 2*n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2013

Mathematica

nn=32; Select[Table[n!Coefficient[Series[Cosh[x]^n, {x, 0, nn}], x^n], {n, 0, nn}], #>0&]
a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Cosh[x]^m, {x, 0, m}]]]; (* Michael Somos, Jul 02 2017 *)

Prog

(PARI) {a(n) = if( n<0, 0, n=2*n; n! * polcoeff( cosh(x + x*O(x^n))^n, n))}; /* Michael Somos, Jul 02 2017 */

Crossrefs

Cf. A085984.

Sequence in context: A000819 A060079 A052502 * A246742 A293950 A104134

Adjacent sequences: A209286 A209287 A209288 * A209290 A209291 A209292

Keyword

nonn

Author

Geoffrey Critzer, Jan 16 2013

Status

approved