A332796 Number of compositions of n^2 into parts >= n.

1, 1, 2, 6, 26, 140, 882, 6349, 51284, 457704, 4459940, 47019819, 532485538, 6438774524, 82710138994, 1123798871990, 16090426592488, 241979954659728, 3811335657375786, 62712512310820402, 1075527196672980525, 19186234784992217621, 355349469934379290700

Offset

0,3

Comments

The n X n Bell square B_n is defined as follows: The first row is all 1's. For each subsequent row, the first number is the last number of the previous row, and the other numbers are the sum of the previous number of this row and the previous number of the previous row. a(n) = B_n[n,n] for n>=1. - Aminos Abijad, Feb 02 2026

Links

Alois P. Heinz, Table of n, a(n) for n = 0..494

Eric Weisstein's World of Mathematics, Bell Triangle.

Formula

a(n) = Sum_{k=0..n} binomial((n-1)*(k+1), n-1-k). - Alois P. Heinz, Feb 27 2026

Example

a(0) = 1: (), the empty composition.
a(1) = 1: 1.
a(2) = 2: 22, 4.
a(3) = 6: 333, 36, 63, 45, 54, 9.
a(4) = 26: 4444, 556, 565, 655, 466, 646, 664, 457, 475, 547, 574, 745, 754, 448, 484, 844, 88, 79, 97, 6(10), (10)6, 5(11), (11)5, 4(12), (12)4, (16).

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k), j=k..n))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..23);
# Alternative:
a:= n-> add(binomial((n-1)*(k+1), n-1-k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 27 2026

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k], {j, k, n}]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

Crossrefs

Cf. A011782, A011971, A103488, A332716, A332721, A332747.

Sequence in context: A030898 A002788 A375652 * A134094 A009575 A263687

Adjacent sequences: A332793 A332794 A332795 * A332797 A332798 A332799

Keyword

nonn

Author

Alois P. Heinz, Feb 24 2020

Status

approved