A131525 Number of degree-2n permutations such that number of cycles of size 2k is odd (or zero) and number of cycles of size 2k-1 is even (or zero), for every k.

1, 2, 13, 371, 17389, 1369057, 168362459, 28396593031, 6237698137129, 1823043651343241, 654314519766396223, 288203550242534470051, 151792464548141462268029, 95104739612472479469277141, 68849533918239714802762113739, 58193958459903387205593351715847

Offset

0,2

Links

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

Formula

E.g.f.: Product(1+sinh(x^(2*k)/(2*k)),k=1..infinity)*Product(cosh(x^(2*k-1)/(2*k-1)),k=1..infinity).

a(n) ~ c * 4^n * n! * (n-1)!, where c = 0.474431... - Vaclav Kotesovec, Jul 21 2019

Example

a(2)=13 because we have (1)(2)(3)(4), six permutations of type (p)(q)(rs) and six permutations of type (pqrs).

Maple

g:=product((1+sinh(x^(2*k)/(2*k)))*cosh(x^(2*k-1)/(2*k-1)), k=1..25): gser:= series(g, x=0, 30): seq(factorial(2*n)*coeff(gser, x, 2*n), n=0..13); # Emeric Deutsch, Sep 04 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+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_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i + j, 2] == 1, multinomial[n, {n - i j} ~Join~ Table[i, {j}]] (i - 1)!^j/j! b[n - i j, i - 1], 0], {j, 0, n/i}]]];
a[n_] := b[2n, 2n];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)

Crossrefs

Cf. A130639, A130644.

Sequence in context: A075620 A336188 A348015 * A366722 A082751 A120935

Adjacent sequences: A131522 A131523 A131524 * A131526 A131527 A131528

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 25 2007

Extensions

More terms from Emeric Deutsch, Sep 04 2007

Status

approved