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

0,3

Comments

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.

Links

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

Formula

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

Example

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

Maple

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

Prog

(PARI)
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)

Crossrefs

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

Sequence in context: A129086 A035052 A148330 * A149876 A165146 A360708

Adjacent sequences: A383272 A383273 A383274 * A383276 A383277 A383278

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 21 2025

Status

approved