A130639 Number of degree-2n permutations without even cycles and such that number of cycles of size 2k-1 is even (or zero) for every k.

1, 1, 1, 41, 1121, 80977, 5073377, 984765497, 131026429249, 45819745767329, 9199822716980033, 5303459200225973833, 1646226697154555000993, 1377111876294420026771441, 574027598120143165861124641, 675477754387947155701063431257, 381022545331716847279242552317057

Offset: 0

Author

Vladeta Jovovic, Aug 11 2007

Keywords

• easy
• nonn

Examples

			a(2)=1 because we have (1)(2)(3)(4).
		

Links

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

Crossrefs

Cf. A060307.

Programs

Maple

g:=product(cosh(x^(2*k-1)/(2*k-1)),k=1..40): gser:=series(g,x=0,35): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Aug 25 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)=1 and irem(j, 2)=0, 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] == 1 && Mod[j, 2] == 0, 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} cosh(x^(2*k-1)/(2*k-1)).

Extensions

More terms from Emeric Deutsch, Aug 25 2007