A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.

1, 1, 4, 86, 2696, 168232, 15948032, 2172623168, 398846422144, 97541017510784, 29909993927387648, 11447388459863715328, 5284740632299379566592, 2927671399386587378671616, 1897593132067741963020476416, 1437515129453860805943287939072

Offset: 0

Author

Vladeta Jovovic, Aug 06 2007

Keywords

• easy
• nonn

Examples

			a(2)=4 because we have (1)(2)(3)(4), (12)(34), (13)(24) and (14)(23).
		

Links

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

Crossrefs

Cf. A055922, A130219, A130220, A130263, A218504, A270597, A270598.

Programs

Maple

g:=product(cosh(x^k/k),k=1..30): gser:=series(g,x=0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # 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(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

nn=26;Select[Range[0,nn]!CoefficientList[Series[Product[Cosh[x^k/k],{k,1,nn}],{x,0,nn}],x],#>0&] (* Geoffrey Critzer, Sep 17 2013 *)

Formula

E.g.f.: Product_{k>0} cosh(x^k/k).

a(n) ~ c * (2*n-1)! / n ~ c * sqrt(Pi) * n^(2*n-3/2) * 2^(2*n) / exp(2*n), where c = A249673 = Product_{k>=1} cosh(1/k) = 2.1164655365... . - Vaclav Kotesovec, Mar 19 2016

Extensions

More terms from Emeric Deutsch, Aug 24 2007