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A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

%I A130275 #11 Dec 22 2016 02:26:40
%S A130275 1,1,2,6,21,105,675,4725,35805,322245,3236625,35602875,425872755,
%T A130275 5536345815,77347084815,1160206272225,18403556596425,312860462139225,
%U A130275 5643104418376425,107218983949152075,2136610763952639975,44868826043005439475,986129980012277775675
%N A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.
%H A130275 Alois P. Heinz, <a href="/A130275/b130275.txt">Table of n, a(n) for n = 0..450</a>
%F A130275 E.g.f.: sqrt((1+x)/(1-x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k))).
%e A130275 a(4)=21 because only the following three degree-4 permutations do not qualify: (12)(34), (13)(24) and (14)(23).
%p A130275 g:=sqrt((1+x)/(1-x))*(product(1+sinh(x^(2*k)/(2*k)),k=1..30)): gser:=series(g, x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # _Emeric Deutsch_, Aug 24 2007
%p A130275 # second Maple program:
%p A130275 with(combinat):
%p A130275 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A130275       `if`(j=0 or irem(i, 2)=1 or irem(j, 2)=1, multinomial(n,
%p A130275        n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
%p A130275     end:
%p A130275 a:= n-> b(n$2):
%p A130275 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 09 2015
%t A130275 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] == 1 || 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 A130275 Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.
%K A130275 easy,nonn
%O A130275 0,3
%A A130275 _Vladeta Jovovic_, Aug 06 2007
%E A130275 More terms from _Emeric Deutsch_, Aug 24 2007