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

0,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A060307.

Sequence in context: A268114 A097713 A009392 * A061812 A009452 A206141

Adjacent sequences: A130645 A130646 A130647 * A130649 A130650 A130651

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 11 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved