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A130639 Number of degree-2n permutations without even cycles and such that number of cycles of size 2k-1 is even (or zero) for every k.

%I A130639 #10 Feb 08 2017 05:32:18
%S A130639 1,1,1,41,1121,80977,5073377,984765497,131026429249,45819745767329,
%T A130639 9199822716980033,5303459200225973833,1646226697154555000993,
%U A130639 1377111876294420026771441,574027598120143165861124641,675477754387947155701063431257,381022545331716847279242552317057
%N A130639 Number of degree-2n permutations without even cycles and such that number of cycles of size 2k-1 is even (or zero) for every k.
%H A130639 Alois P. Heinz, <a href="/A130639/b130639.txt">Table of n, a(n) for n = 0..220</a>
%F A130639 E.g.f.: Product_{k>0} cosh(x^(2*k-1)/(2*k-1)).
%e A130639 a(2)=1 because we have (1)(2)(3)(4).
%p A130639 g:=product(cosh(x^(2*k-1)/(2*k-1)),k=1..40): gser:=series(g,x=0,35): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # _Emeric Deutsch_, Aug 25 2007
%p A130639 # second Maple program:
%p A130639 with(combinat):
%p A130639 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130639       `if`(j=0 or irem(i, 2)=1 and irem(j, 2)=0, multinomial(n,
%p A130639        n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130639     end:
%p A130639 a:= n-> b(2*n$2):
%p A130639 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 09 2015
%t A130639 multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 && 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_, Feb 08 2017, after _Alois P. Heinz_ *)
%Y A130639 Cf. A060307.
%K A130639 easy,nonn
%O A130639 0,4
%A A130639 _Vladeta Jovovic_, Aug 11 2007
%E A130639 More terms from _Emeric Deutsch_, Aug 25 2007