A213048 Number of preferential arrangements of n labeled elements with repetitions allowed.

1, 1, 5, 31, 229, 1981, 19775, 224589, 2864901, 40591255, 632760105, 10765616885, 198543617119, 3945765358653, 84070841065937, 1911864488674531, 46222718892288645, 1183919151676806177, 32025836905529003471, 912372909851608715945, 27304698509111141688969

Offset: 0

Author

Thomas Wieder, Jun 03 2012

Keywords

• nonn

Examples

			For n=2 the a(2) = 5 solutions are (1,2), (1|2), (2|1), (1|1), (2|2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A000670.

Programs

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, add(binomial(n+k-1, k)*a(n-k), k=1..n))
    end:
seq(a(n), n=0..25);

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n+k-1, k] a[n-k], {k, 1, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 21 2020 *)

Formula

a(n) = Sum_{k=1..n} C(n+k-1,k)*a(n-k) for n>0, a(0) = 1.

a(n) = Sum_c(n) C(n+k1-1,k1) C(n-k1+k2-1,k2) C(n-k1-k2+k3-1,k3) ..., where Sum_c(n) denotes the sum over all compositions (ordered partitions) of n = k1 + k2 + ... .

a(n) ~ c * n! * n^(log(2)) / (log(2))^n, where c = 0.9387523255426859866752735339706007723805611... . - Vaclav Kotesovec, May 03 2015