A101109 Number of sets of lists (sequences) of n labeled elements with k=3 elements per list.

1, 0, 0, 6, 0, 0, 360, 0, 0, 60480, 0, 0, 19958400, 0, 0, 10897286400, 0, 0, 8892185702400, 0, 0, 10137091700736000, 0, 0, 15388105201717248000, 0, 0, 30006805143348633600000, 0, 0, 73096577329197271449600000

Offset

0,4

Comments

The (labeled) case for k=2 is A067994, the Hermite numbers. The (labeled) case for k>=1 is A000262, Number of "sets of lists".

Links

Alois P. Heinz, Table of n, a(n) for n = 0..582

Formula

E.g.f.: exp(z^3).

a(0) = 1, a(1) = 0, a(2) = 0, (-n-3)*a(n+3)+3*a(n).

a(n) = n!/(n/3)!, if 3 divides n, 0 otherwise. - Mitch Harris, Jan 19 2006

Example

Let Z[i] denote the i-th labeled element. Then a(3) = 6 with the following six sets:
Set(Sequence(Z[3],Z[1],Z[2])), Set(Sequence(Z[2],Z[1],Z[3])), Set(Sequence(Z[3],Z[2],Z[1])), Set(Sequence(Z[2],Z[3],Z[1])), Set(Sequence(Z[1],Z[3],Z[2])), Set(Sequence(Z[1],Z[2],Z[3])).

Maple

A101109 := n -> n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n, 3) = 0], [0, irem(n-1, 3) = 0], [0, irem(n-2, 3) = 0]); [ seq(n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n, 3) = 0], [0, irem(n-1, 3) = 0], [0, irem(n-2, 3) = 0]), n=0..30) ];
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
       j!*binomial(n-1, j-1), j=`if`(n>2, 3, [][])))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 10 2016

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x^3], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 16 2013 *)

Prog

(Sage)
def A101109(n) : return factorial(n)/factorial(n/3) if n%3 == 0 else 0
[A101109(n) for n in (0..30)] # Peter Luschny, Jul 12 2012

Crossrefs

Cf. A000262, A067994.

Sequence in context: A358515 A230787 A360424 * A353224 A293526 A293568

Adjacent sequences: A101106 A101107 A101108 * A101110 A101111 A101112

Keyword

nonn

Author

Thomas Wieder, Dec 01 2004

Status

approved