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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A000670.

Sequence in context: A001910 A052773 A062147 * A349535 A069321 A211179

Adjacent sequences: A213045 A213046 A213047 * A213049 A213050 A213051

Keyword

nonn

Author

Thomas Wieder, Jun 03 2012

Status

approved