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

0,3

Links

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

Formula

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

Maple

b:= proc(n, i, p, k) option remember;
      `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))))
    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 *)

Crossrefs

Main diagonal of A261835.

Sequence in context: A370458 A037105 A196137 * A352756 A059624 A278621

Adjacent sequences: A261834 A261835 A261836 * A261838 A261839 A261840

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2015

Status

approved