A130648 Number of degree-n permutations without even cycles and such that number of cycles of size 2k-1 is odd (or zero) for every k.

1, 1, 0, 3, 8, 25, 184, 721, 9904, 66753, 691088, 5973121, 84925048, 940427137, 12801319816, 186556383105, 3174772979936, 48489077948161, 842173637012896, 15359492773456129, 316965131969908072, 6368424993521096961, 135098381153771956952, 2980219360336428021505

Offset: 0

Author

Vladeta Jovovic, Aug 11 2007

Keywords

• easy
• nonn

Examples

			a(3)=3 because we have (1)(2)(3), (123) and (132).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A060307.

Programs

Maple

g:=product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30): gser:=series(g,x=0,27): seq(factorial(n)*coeff(gser,x,n),n=0..24); # 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)=1 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(n$2):
seq(a(n), n=0..30);  # 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] == 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[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

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

Extensions

More terms from Emeric Deutsch, Aug 24 2007