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

1, 1, 1, 6, 17, 100, 529, 3766, 31121, 276984, 2755553, 29665306, 364627801, 4639937380, 64952094401, 973467571350, 15750475301921, 264870218828656, 4759194994114369, 90124395399063730, 1812001488739061417, 37956199941196210716, 832297726351555617569

Offset: 0

Author

Vladeta Jovovic, Aug 06 2007

Keywords

• easy
• nonn

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

Programs

Maple

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
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

Mathematica

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 *)

Formula

E.g.f.: 1/sqrt(1-x^2)*Product_{k>0} (1+sinh(x^(2*k-1)/(2*k-1))).

Extensions

More terms from Emeric Deutsch, Aug 24 2007