A052565 E.g.f. (1+x^3-x^4)/(1-x).

1, 1, 2, 12, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Number of partitions of n-set into "lists", in which every even list appears an odd number of times, cf. A000262. - Alois P. Heinz, May 10 2016

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 507

Crossrefs

Cf. A000262, A102760.

Programs

Maple

spec := [S,{S=Union(Sequence(Z),Prod(Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i::even and j::even, 0, b(n-i*j, i-1)*
      multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Mathematica

a[n_] := If[n <= 3, {1, 1, 2, 12}[[n+1]], n!];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 10 2020 *)

Formula

E.g.f.: (-1+x^4-x^3)/(-1+x).

Recurrence: {a(1)=1, a(0)=1, (-1-n)*a(n)+a(n+1)=0, a(2)=2, a(4)=24, a(3)=12}.

a(n) = n! for n>3.