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A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

%I A130276 #19 Dec 22 2016 03:41:31
%S A130276 1,2,16,416,20224,1645312,196388864,33279311872,7427338829824,
%T A130276 2151276556845056,771086221948223488,340572557390992900096,
%U A130276 179222835344084459061248,112158801651454395931426816,81399358513573250066141937664,68530340884909785149816189222912
%N A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.
%H A130276 Alois P. Heinz, <a href="/A130276/b130276.txt">Table of n, a(n) for n = 0..220</a>
%F A130276 E.g.f. with interleaved zeros: 1/sqrt(1-x^2)*Product_{k>=1} cosh(x^(2*k-1)/(2*k-1)). - _Geoffrey Critzer_, Jan 02 2011
%e A130276 a(2)=16 because there are 8 permutations that do not qualify: (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3) and (143)(2).
%p A130276 g:=(product(cosh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x= 0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # _Emeric Deutsch_, Aug 24 2007
%p A130276 # second Maple program:
%p A130276 with(combinat):
%p A130276 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130276       `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=0, multinomial(n,
%p A130276        n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130276     end:
%p A130276 a:= n-> b(2*n$2):
%p A130276 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 09 2015
%t A130276 multinomial[n_, k_List] := 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] == 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_, Dec 22 2016, after _Alois P. Heinz_ *)
%o A130276 (PARI) N=31; x='x+O('x^N);
%o A130276 v0=Vec(serlaplace(1/sqrt(1-x^2)*prod(k=1,N, cosh(x^(2*k-1)/(2*k-1)))));
%o A130276 vector(#v0\2,n,v0[2*n-1]) \\ _Joerg Arndt_, Jan 03 2011
%Y A130276 Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.
%K A130276 easy,nonn
%O A130276 0,2
%A A130276 _Vladeta Jovovic_, Aug 06 2007
%E A130276 More terms from _Emeric Deutsch_, Aug 24 2007