A382991 Number of compositions of n such that any part 1 at position k can be k different colors.

1, 1, 3, 10, 40, 193, 1110, 7473, 57821, 505945, 4940354, 53248874, 627848885, 8037734930, 111017325473, 1645384681765, 26044845197881, 438499277779636, 7824114643731522, 147476551001255125, 2928074880767254238, 61078483577649288463, 1335438738400978511877

Offset: 0

Author

John Tyler Rascoe, Apr 11 2025

Keywords

• nonn
• easy

Examples

			a(3) = 10 counts: (3), (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. A008275, A011782, A088305, A238351, A240736, A382992.

Programs

Maple

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

PARI

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

Formula

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