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A382991 Number of compositions of n such that any part 1 at position k can be k different colors.

%I A382991 #14 Apr 24 2025 08:01:52
%S A382991 1,1,3,10,40,193,1110,7473,57821,505945,4940354,53248874,627848885,
%T A382991 8037734930,111017325473,1645384681765,26044845197881,438499277779636,
%U A382991 7824114643731522,147476551001255125,2928074880767254238,61078483577649288463,1335438738400978511877
%N A382991 Number of compositions of n such that any part 1 at position k can be k different colors.
%H A382991 Alois P. Heinz, <a href="/A382991/b382991.txt">Table of n, a(n) for n = 0..450</a>
%F A382991 G.f.: 1 + Sum_{i>0} Product_{j=1..i} ( j*x + x^2/(1-x) ).
%e A382991 a(3) = 10 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_a,1_a,1_a), (1_a,1_a,1_b), (1_a,1_a,1_c), (1_a,1_b,1_a), (1_a,1_b,1_b), (1_a,1_b,1_c).
%p A382991 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A382991       add(b(n-j, i+1)*`if`(j=1, i, 1), j=1..n))
%p A382991     end:
%p A382991 a:= n-> b(n, 1):
%p A382991 seq(a(n), n=0..22);  # _Alois P. Heinz_, Apr 23 2025
%o A382991 (PARI)
%o A382991 A_x(N) = {my(x='x+O('x^N)); Vec(1+ sum(i=1,N, prod(j=1,i, j*x + x^2/(1-x))))}
%o A382991 A_x(30)
%Y A382991 Cf. A008275, A011782, A088305, A238351, A240736, A382992.
%K A382991 nonn,easy
%O A382991 0,3
%A A382991 _John Tyler Rascoe_, Apr 11 2025