A382820 Number of columns in all n-compositions of n.

1, 11, 163, 3019, 67251, 1753877, 52468711, 1772042699, 66708748963, 2770212058261, 125812351808551, 6203908746628501, 330108021642012407, 18853083403505443593, 1150352428059538611663, 74685045367715777653195, 5140745255774277374241411, 373950591013899715795929605

Offset: 1

Author

John Tyler Rascoe, Apr 05 2025

Keywords

• nonn
• easy

Comments

A k-composition of n is a rectangular array of nonnegative integers with k rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

Crossrefs

C.f. A001792, A145839, A181289, A181290, (main diagonal of A382818).

Programs

PARI

A382818_Column(k,N) = {my(x='x+O('x^N)); Vec(-(((1 - x)^k - 1)*(1 - x)^k)/( ((1 - x)^k - 1) + (1 - x)^k)^2)}
A382820(n)= {A382818_Column(n,n+1)[n]}

Formula

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