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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.

%I A131525 #14 Nov 19 2020 07:54:50
%S A131525 1,2,13,371,17389,1369057,168362459,28396593031,6237698137129,
%T A131525 1823043651343241,654314519766396223,288203550242534470051,
%U A131525 151792464548141462268029,95104739612472479469277141,68849533918239714802762113739,58193958459903387205593351715847
%N 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.
%H A131525 Alois P. Heinz, <a href="/A131525/b131525.txt">Table of n, a(n) for n = 0..220</a>
%F A131525 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).
%F A131525 a(n) ~ c * 4^n * n! * (n-1)!, where c = 0.474431... - _Vaclav Kotesovec_, Jul 21 2019
%e A131525 a(2)=13 because we have (1)(2)(3)(4), six permutations of type (p)(q)(rs) and six permutations of type (pqrs).
%p A131525 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
%p A131525 # second Maple program:
%p A131525 with(combinat):
%p A131525 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A131525       `if`(j=0 or irem(i+j, 2)=1, multinomial(n, n-i*j, i$j)*
%p A131525        (i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A131525     end:
%p A131525 a:= n-> b(2*n$2):
%p A131525 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 09 2015
%t A131525 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A131525 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}]]];
%t A131525 a[n_] := b[2n, 2n];
%t A131525 a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)
%Y A131525 Cf. A130639, A130644.
%K A131525 easy,nonn
%O A131525 0,2
%A A131525 _Vladeta Jovovic_, Aug 25 2007
%E A131525 More terms from _Emeric Deutsch_, Sep 04 2007