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A383175 Number of compositions of n such that any fixed point k can be k different colors.

%I A383175 #13 Apr 21 2025 16:27:11
%S A383175 1,1,2,5,10,22,48,101,213,450,945,1961,4064,8385,17242,35332,72141,
%T A383175 146924,298552,605377,1225277,2475912,4995754,10067848,20267680,
%U A383175 40762951,81916919,164504411,330155437,662265817,1327860471,2661376529,5332341881,10680912173
%N A383175 Number of compositions of n such that any fixed point k can be k different colors.
%H A383175 Alois P. Heinz, <a href="/A383175/b383175.txt">Table of n, a(n) for n = 0..2000</a>
%F A383175 G.f.: 1 + Sum_{i>0} Product_{j=1..i} ( j*x^j - x^j + x/(1-x) ).
%e A383175 a(3) = 5 counts: (3), (2,1), (1_a,2_a), (1_a,2_b), (1_a,1,1).
%p A383175 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A383175      `if`(n<=i+j, ceil(2^(n-j-1)), b(n-j, i+1))*
%p A383175      `if`(i=j, j, 1), j=1..n))
%p A383175     end:
%p A383175 a:= n-> b(n, 1):
%p A383175 seq(a(n), n=0..33);  # _Alois P. Heinz_, Apr 18 2025
%o A383175 (PARI)
%o A383175 A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N, prod(j=1,i, j*x^j-x^j+x/(1-x))))}
%o A383175 A_x(30)
%Y A383175 Cf. A011782, A088305, A238349, A238350, A238351, A335713, A352512.
%K A383175 nonn,easy
%O A383175 0,3
%A A383175 _John Tyler Rascoe_, Apr 18 2025