A383101 -id:A383101 - cp's OEIS frontend

A383275 Number of compositions of n such that any part 1 can be k different colors where k is the current record having appeared in the composition.

1, 1, 2, 5, 14, 42, 134, 454, 1634, 6245, 25321, 108779, 494443, 2374288, 12024257, 64100444, 358948674, 2106756217, 12931155910, 82823317389, 552400947902, 3829070637080, 27534807426150, 205066734143893, 1579309451332366, 12559941159979791, 103013928588389695

Offset: 0

John Tyler Rascoe, Apr 21 2025

A record in a composition is a part that is greater than all parts before it, reading left to right. The first part of any nonempty composition is considered a record. A part 1 can be a record, iff it is the first part of a composition.

			a(3) = 5: (3), (1_a,2), (2,1_a), (2,1_b), (1_a,1_a,1_a).

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A000108, A011782, A088305, A382312, A382991, A383101, A383175.

b:= proc(n, m) option remember; `if`(n=0, 1, add(
      b(n-j, max(j, m))*`if`(j=1, m, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 23 2025

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

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

cp's OEIS Frontend

A383275 Number of compositions of n such that any part 1 can be k different colors where k is the current record having appeared in the composition.

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