A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.

1, 1, 7, 73, 1031, 18501, 403495, 10366833, 306717703, 10271072557, 384058268507, 15861842372465, 717135437119271, 35228475333207937, 1868440035684996207, 106412817671933423073, 6477200889282232394759, 419626639092214594301373, 28829330550533269570699411

Offset: 0

Author

Alois P. Heinz, Aug 31 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..365

Crossrefs

Main diagonal of A261780.

Cf. A209668.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - j, k]*Binomial[j + k - 1, k - 1], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2017, translated from Maple *)

Formula

a(n) = A261780(n,n).

a(n) = [x^n] (1-x)^n/(2*(1-x)^n-1).

a(n) ~ n^n / (sqrt(2) * (log(2))^(n+1)). - Vaclav Kotesovec, Sep 21 2019

a(n) = Sum_{k>=1} (1/2)^k * binomial(k*n-1,n). - Seiichi Manyama, Aug 06 2024