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

0,3

Links

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

Formula

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

Example

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).

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

Prog

(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)

Crossrefs

Cf. A008275, A011782, A088305, A238351, A240736, A382992.

Sequence in context: A231531 A136128 A089902 * A093133 A030817 A030845

Adjacent sequences: A382988 A382989 A382990 * A382992 A382993 A382994

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 11 2025

Status

approved