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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.

%I A383101 #16 Apr 24 2025 09:34:38
%S A383101 1,1,2,6,21,77,294,1178,4978,22191,104146,513385,2653003,14349804,
%T A383101 81125023,478686413,2943737942,18838530436,125268429098,864256288435,
%U A383101 6177766228172,45689641883377,349173454108407,2754058599745239,22393206702946457,187501022603071090
%N 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.
%H A383101 Alois P. Heinz, <a href="/A383101/b383101.txt">Table of n, a(n) for n = 0..500</a>
%F A383101 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))).
%e A383101 a(3) = 6 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_b,2), (1_a,1_a,1_a).
%p A383101 b:= proc(n, p, m) option remember; binomial(n+p, n)*
%p A383101       m^n+add(b(n-j, p+1, max(m, j)), j=2..n)
%p A383101     end:
%p A383101 a:= n-> b(n, 0, 1):
%p A383101 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 23 2025
%o A383101 (PARI)
%o A383101 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)))))}
%o A383101 A_x(30)
%Y A383101 Cf. A011782, A088305, A105422, A171632, A363262, A382991, A382992.
%K A383101 nonn,easy
%O A383101 0,3
%A A383101 _John Tyler Rascoe_, Apr 16 2025