A130644 Number of degree-2n permutations without odd cycles and such that number of cycles of size 2k is odd (or zero) for every k.

1, 1, 6, 225, 8400, 760725, 91725480, 15563633085, 3381661483200, 1015992072520425, 360153767651277600, 160068908768727783825, 84298688029883001074400, 53051020433282263735468125, 38316864396320965168213500000, 32660810942813910822645908353125

Offset: 0

Author

Vladeta Jovovic, Aug 11 2007

Keywords

• easy
• nonn

Examples

			a(2)=6 because we have (1234),(1243),(1324),(1342),(1423) and (1432).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..220

Crossrefs

Cf. A060307.

Programs

Maple

g:=product(1+sinh(x^(2*k)/(2*k)),k=1..50): gser:=series(g,x=0,44): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 and irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 && Mod[j, 2] == 1, multinomial[n, Join[{n-i*j}, Array[i&, j]]]*(i-1)!^j/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

E.g.f.: Product_{k>0} (1+sinh(x^(2*k)/(2*k))).

Extensions

More terms from Emeric Deutsch, Aug 24 2007