A382992 Number of compositions of n that have at least 1 part equal to 1 and any part 1 at position k can be k different colors.

0, 1, 2, 9, 38, 190, 1105, 7465, 57808, 505924, 4940320, 53248819, 627848796, 8037734786, 111017325240, 1645384681388, 26044845197271, 438499277778649, 7824114643729925, 147476551001252541, 2928074880767250057, 61078483577649281698, 1335438738400978500931

Offset: 0

Author

John Tyler Rascoe, Apr 11 2025

Keywords

• nonn
• easy

Examples

			a(3) = 9 counts: (2, 1_a), (2, 1_b), (1_a, 2), (1_a, 1_a, 1_a), (1_a, 1_a, 1_b), (1_a, 1_a, 1_c), (1_a, 1_b, 1_a), (1_a, 1_b, 1_b), (1_a, 1_b, 1_c).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000045, A008275, A011782, A088305, A238351, A240736, A382991.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t, add(
      b(n-j, i+1, `if`(j=1, 1, t))*`if`(j=1, i, 1), j=1..n))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 23 2025

PARI

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

Formula

G.f.: -x^2/(1-x-x^2) + Sum_{i>0} Product_{j=1..i} ( j*x + x^2/(1-x) ).

a(n) = A382991(n) - A000045(n-1).

Extensions

Edited by Alois P. Heinz, Apr 23 2025