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A130644 Number of degree-2n permutations without odd cycles and such that number of cycles of size 2k is odd (or zero) for every k.

%I A130644 #11 Feb 08 2017 05:32:09
%S A130644 1,1,6,225,8400,760725,91725480,15563633085,3381661483200,
%T A130644 1015992072520425,360153767651277600,160068908768727783825,
%U A130644 84298688029883001074400,53051020433282263735468125,38316864396320965168213500000,32660810942813910822645908353125
%N A130644 Number of degree-2n permutations without odd cycles and such that number of cycles of size 2k is odd (or zero) for every k.
%H A130644 Alois P. Heinz, <a href="/A130644/b130644.txt">Table of n, a(n) for n = 0..220</a>
%F A130644 E.g.f.: Product_{k>0} (1+sinh(x^(2*k)/(2*k))).
%e A130644 a(2)=6 because we have (1234),(1243),(1324),(1342),(1423) and (1432).
%p A130644 g:=product(1+sinh(x^(2*k)/(2*k)),k=1..50): gser:=series(g,x=0,44): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # _Emeric Deutsch_, Aug 24 2007
%p A130644 # second Maple program:
%p A130644 with(combinat):
%p A130644 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130644       `if`(j=0 or irem(i, 2)=0 and irem(j, 2)=1, multinomial(n,
%p A130644        n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130644     end:
%p A130644 a:= n-> b(2*n$2):
%p A130644 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 09 2015
%t A130644 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] == 0 && Mod[j, 2] == 1, 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 A130644 Cf. A060307.
%K A130644 easy,nonn
%O A130644 0,3
%A A130644 _Vladeta Jovovic_, Aug 11 2007
%E A130644 More terms from _Emeric Deutsch_, Aug 24 2007