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A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

%I A130278 #11 Dec 22 2016 03:41:23
%S A130278 1,1,1,6,17,100,529,3766,31121,276984,2755553,29665306,364627801,
%T A130278 4639937380,64952094401,973467571350,15750475301921,264870218828656,
%U A130278 4759194994114369,90124395399063730,1812001488739061417,37956199941196210716,832297726351555617569
%N A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.
%H A130278 Alois P. Heinz, <a href="/A130278/b130278.txt">Table of n, a(n) for n = 0..450</a>
%F A130278 E.g.f.: 1/sqrt(1-x^2)*Product_{k>0} (1+sinh(x^(2*k-1)/(2*k-1))).
%e A130278 a(4)=17 because only the following 7 permutations do not qualify: (1)(2)(3)(4), (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4) and (14)(2)(3).
%p A130278 g:=(product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x =0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # _Emeric Deutsch_, Aug 24 2007
%p A130278 # second Maple program:
%p A130278 with(combinat):
%p A130278 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130278       `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=1, multinomial(n,
%p A130278        n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130278     end:
%p A130278 a:= n-> b(n$2):
%p A130278 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 09 2015
%t A130278 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] == 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[n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A130278 Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.
%K A130278 easy,nonn
%O A130278 0,4
%A A130278 _Vladeta Jovovic_, Aug 06 2007
%E A130278 More terms from _Emeric Deutsch_, Aug 24 2007