A383101 Number of compositions of n such that any part 1 can be m different colors where m is the largest part of the composition.

1, 1, 2, 6, 21, 77, 294, 1178, 4978, 22191, 104146, 513385, 2653003, 14349804, 81125023, 478686413, 2943737942, 18838530436, 125268429098, 864256288435, 6177766228172, 45689641883377, 349173454108407, 2754058599745239, 22393206702946457, 187501022603071090

Offset: 0

Author

John Tyler Rascoe, Apr 16 2025

Keywords

• nonn
• easy

Examples

			a(3) = 6 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_b,2), (1_a,1_a,1_a).
		

Links

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

Crossrefs

Cf. A011782, A088305, A105422, A171632, A363262, A382991, A382992.

Programs

Maple

b:= proc(n, p, m) option remember; binomial(n+p, n)*
      m^n+add(b(n-j, p+1, max(m, j)), j=2..n)
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 23 2025

PARI

A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(m=1,N, x^m/((1-m*x-(x^2-x^m)/(1-x))*(1-m*x-(x^2-x^(m+1))/(1-x)))))}
A_x(30)

Formula

G.f.: 1 + Sum_{m>0} x^m/((1 - m*x - (x^2 - x^m)/(1 - x)) * (1 - m*x - (x^2 - x^(m+1))/(1 - x))).