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A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.

%I A130268 #22 Mar 20 2016 11:46:33
%S A130268 1,1,4,86,2696,168232,15948032,2172623168,398846422144,97541017510784,
%T A130268 29909993927387648,11447388459863715328,5284740632299379566592,
%U A130268 2927671399386587378671616,1897593132067741963020476416,1437515129453860805943287939072
%N A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.
%H A130268 Alois P. Heinz, <a href="/A130268/b130268.txt">Table of n, a(n) for n = 0..220</a>
%F A130268 E.g.f.: Product_{k>0} cosh(x^k/k).
%F A130268 a(n) ~ c * (2*n-1)! / n ~ c * sqrt(Pi) * n^(2*n-3/2) * 2^(2*n) / exp(2*n), where c = A249673 = Product_{k>=1} cosh(1/k) = 2.1164655365... . - _Vaclav Kotesovec_, Mar 19 2016
%e A130268 a(2)=4 because we have (1)(2)(3)(4), (12)(34), (13)(24) and (14)(23).
%p A130268 g:=product(cosh(x^k/k),k=1..30): gser:=series(g,x=0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # _Emeric Deutsch_, Aug 24 2007
%p A130268 # second Maple program:
%p A130268 with(combinat):
%p A130268 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A130268       add(`if`(j=0 or irem(j, 2)=0, multinomial(n, n-i*j, i$j)
%p A130268        *(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130268     end:
%p A130268 a:= n-> b(2*n$2):
%p A130268 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 09 2015
%t A130268 nn=26;Select[Range[0,nn]!CoefficientList[Series[Product[Cosh[x^k/k],{k,1,nn}],{x,0,nn}],x],#>0&] (* _Geoffrey Critzer_, Sep 17 2013 *)
%Y A130268 Cf. A055922, A130219, A130220, A130263, A218504, A270597, A270598.
%K A130268 easy,nonn
%O A130268 0,3
%A A130268 _Vladeta Jovovic_, Aug 06 2007
%E A130268 More terms from _Emeric Deutsch_, Aug 24 2007