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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A055591 A055764 A163279 * A293318 A204460 A162086

Adjacent sequences: A130265 A130266 A130267 * A130269 A130270 A130271

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 06 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved