A261837 Number of compositions of n into distinct parts 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, 3, 46, 195, 1876, 51114, 322764, 3644355, 43916950, 2427338628, 18277511616, 272107762602, 3507931293608, 62485721142820, 5810222040368296, 53025343448015811, 913540133071336044, 13871534219465464002, 253750203721349071650, 5307815745011707670820

Offset: 0

Author

Alois P. Heinz, Sep 02 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Main diagonal of A261835.

Programs

Maple

b:= proc(n, i, p, k) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> b(n$2, 0, n):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, p_, k_] := b[n, i, p, k] =
     If[i (i + 1)/2 < n, 0, If[n == 0, p!, b[n, i - 1, p, k] +
     If[i > n, 0, b[n - i, i - 1, p + 1, k]*Binomial[i + k - 1, k - 1]]]];
a[n_] := b[n, n, 0, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

Formula

a(n) = A261835(n,n).