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A131526 Number of degree-n permutations such that number of cycles of size 2k is even (or zero) and number of cycles of size 2k-1 is odd (or zero), for every k.

%I A131526 #13 Nov 19 2020 12:01:33
%S A131526 1,1,0,3,11,40,184,1036,12949,88488,807008,7362586,113572183,
%T A131526 1238477032,15630890560,228998728050,4141605806441,62222251093216,
%U A131526 1030119451142656,19050688698470434,412037845709792107,8102391640556570616,165794307361686866432
%N A131526 Number of degree-n permutations such that number of cycles of size 2k is even (or zero) and number of cycles of size 2k-1 is odd (or zero), for every k.
%H A131526 Alois P. Heinz, <a href="/A131526/b131526.txt">Table of n, a(n) for n = 0..450</a>
%F A131526 E.g.f.: Product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..infinity) *Product(cosh(x^(2*k)/(2*k)), k=1..infinity).
%e A131526 a(4)=11 because we have (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3), (143)(2), (12)(34), (13)(24) and (14)(23).
%p A131526 g:=(product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..40))*(product(cosh(x^(2*k)/(2*k)), k=1..40)): gser:=series(g,x=0,25); seq(factorial(n)*coeff(gser,x,n),n=0..21); # _Emeric Deutsch_, Aug 28 2007
%p A131526 # second Maple program:
%p A131526 with(combinat):
%p A131526 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A131526       `if`(j=0 or irem(i+j, 2)=0, multinomial(n, n-i*j, i$j)*
%p A131526        (i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A131526     end:
%p A131526 a:= n-> b(n$2):
%p A131526 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 09 2015
%t A131526 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A131526 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i + j, 2] == 0, 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 A131526 a[n_] := b[n, n];
%t A131526 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)
%Y A131526 Cf. A060307, A130648.
%K A131526 easy,nonn
%O A131526 0,4
%A A131526 _Vladeta Jovovic_, Aug 25 2007
%E A131526 More terms from _Emeric Deutsch_, Aug 28 2007