A131526 Number of degree-n permutations such that number of cycles of size 2k is even (or zero) and number of cycles of size 2k-1 is odd (or zero), for every k.

1, 1, 0, 3, 11, 40, 184, 1036, 12949, 88488, 807008, 7362586, 113572183, 1238477032, 15630890560, 228998728050, 4141605806441, 62222251093216, 1030119451142656, 19050688698470434, 412037845709792107, 8102391640556570616, 165794307361686866432

Offset

0,4

Links

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

Formula

E.g.f.: Product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..infinity) *Product(cosh(x^(2*k)/(2*k)), k=1..infinity).

Example

a(4)=11 because we have (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3), (143)(2), (12)(34), (13)(24) and (14)(23).

Maple

g:=(product(1+sinh(x^(2*k-1)/(2*k-1)), k=1..40))*(product(cosh(x^(2*k)/(2*k)), k=1..40)): gser:=series(g, x=0, 25); seq(factorial(n)*coeff(gser, x, n), n=0..21); # Emeric Deutsch, Aug 28 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+j, 2)=0, 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 + j, 2] == 0, multinomial[n, {n - i j} ~Join~ Table[i, {j}]] (i - 1)!^j/j! b[n - i j, i - 1], 0], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)

Crossrefs

Cf. A060307, A130648.

Sequence in context: A242467 A149064 A149065 * A329261 A073622 A351428

Adjacent sequences: A131523 A131524 A131525 * A131527 A131528 A131529

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 25 2007

Extensions

More terms from Emeric Deutsch, Aug 28 2007

Status

approved