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

0,3

Links

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

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))).

Example

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

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

Prog

(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)

Crossrefs

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

Sequence in context: A375443 A063023 A150188 * A150189 A144169 A355152

Adjacent sequences: A383098 A383099 A383100 * A383102 A383103 A383104

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 16 2025

Status

approved