A382992 -id:A382992 - cp's OEIS frontend

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

John Tyler Rascoe, Apr 11 2025

			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).

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

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

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

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)

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

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.

Original entry on oeis.org

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

John Tyler Rascoe, Apr 16 2025

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

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

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

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

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)

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))).

cp's OEIS Frontend

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

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