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

0,4

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

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

Sequence in context: A219294 A338747 A154494 * A024080 A099436 A209482

Adjacent sequences: A130275 A130276 A130277 * A130279 A130280 A130281

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 06 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved