A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.
1, 2, 16, 416, 20224, 1645312, 196388864, 33279311872, 7427338829824, 2151276556845056, 771086221948223488, 340572557390992900096, 179222835344084459061248, 112158801651454395931426816, 81399358513573250066141937664, 68530340884909785149816189222912
Offset: 0
Examples
a(2)=16 because there are 8 permutations that do not qualify: (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3) and (143)(2).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..220
Programs
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Maple
g:=(product(cosh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): 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(i, 2)=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
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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, 2] == 0 || Mod[j, 2] == 0, 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[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *) -
PARI
N=31; x='x+O('x^N); v0=Vec(serlaplace(1/sqrt(1-x^2)*prod(k=1,N, cosh(x^(2*k-1)/(2*k-1))))); vector(#v0\2,n,v0[2*n-1]) \\ Joerg Arndt, Jan 03 2011
Formula
E.g.f. with interleaved zeros: 1/sqrt(1-x^2)*Product_{k>=1} cosh(x^(2*k-1)/(2*k-1)). - Geoffrey Critzer, Jan 02 2011
Extensions
More terms from Emeric Deutsch, Aug 24 2007