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.

%I A383275 #18 Jun 11 2025 04:01:14
%S A383275 1,1,2,5,14,42,134,454,1634,6245,25321,108779,494443,2374288,12024257,
%T A383275 64100444,358948674,2106756217,12931155910,82823317389,552400947902,
%U A383275 3829070637080,27534807426150,205066734143893,1579309451332366,12559941159979791,103013928588389695
%N 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.
%C A383275 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.
%H A383275 Alois P. Heinz, <a href="/A383275/b383275.txt">Table of n, a(n) for n = 0..600</a>
%F A383275 G.f.: Product_{i>0} 1 + x^i/(1 - i*x - (x^2 - x^(i+1))/(1-x)).
%e A383275 a(3) = 5: (3), (1_a,2), (2,1_a), (2,1_b), (1_a,1_a,1_a).
%p A383275 b:= proc(n, m) option remember; `if`(n=0, 1, add(
%p A383275       b(n-j, max(j, m))*`if`(j=1, m, 1), j=1..n))
%p A383275     end:
%p A383275 a:= n-> b(n, 1):
%p A383275 seq(a(n), n=0..26);  # _Alois P. Heinz_, Apr 23 2025
%o A383275 (PARI)
%o A383275 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))))}
%o A383275 A_x(30)
%Y A383275 Cf. A000108, A011782, A088305, A382312, A382991, A383101, A383175.
%K A383275 nonn,easy
%O A383275 0,3
%A A383275 _John Tyler Rascoe_, Apr 21 2025