A052565 - OEIS

%I #22 Nov 10 2020 08:20:28

%S 1,1,2,12,24,120,720,5040,40320,362880,3628800,39916800,479001600,

%T 6227020800,87178291200,1307674368000,20922789888000,355687428096000,

%U 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000

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

%C 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

%H Alois P. Heinz, <a href="/A052565/b052565.txt">Table of n, a(n) for n = 0..450</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=507">Encyclopedia of Combinatorial Structures 507</a>

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

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

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

%p spec := [S,{S=Union(Sequence(Z),Prod(Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(`if`(i::even and j::even, 0, b(n-i*j, i-1)*

%p       multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, May 10 2016

%t a[n_] := If[n <= 3, {1, 1, 2, 12}[[n+1]], n!];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 10 2020 *)

%Y Cf. A000262, A102760.

%K easy,nonn

%O 0,3

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