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
Examples
a(2)=13 because we have (1)(2)(3)(4), six permutations of type (p)(q)(rs) and six permutations of type (pqrs).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..220
Programs
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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
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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 + 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 *)
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
Extensions
More terms from Emeric Deutsch, Sep 04 2007