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A327847 Number of colored compositions of n using all colors of a 2-set such that all parts have different color patterns and the patterns for parts i have i colors in (weakly) increasing order.

%I A327847 #6 Sep 28 2019 14:33:43
%S A327847 0,0,3,10,39,100,303,782,2009,5388,12839,32658,79673,191500,459745,
%T A327847 1090622,2569597,5932304,13906405,31740090,72160311,164296384,
%U A327847 369673125,827781598,1834885695,4076433368,8966677703,19639987986,42751905899,92703023676,200647122467
%N A327847 Number of colored compositions of n using all colors of a 2-set such that all parts have different color patterns and the patterns for parts i have i colors in (weakly) increasing order.
%H A327847 Alois P. Heinz, <a href="/A327847/b327847.txt">Table of n, a(n) for n = 0..800</a>
%e A327847 a(3) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
%p A327847 b:= proc(n, i, k, p) option remember; `if`(n=0, p!,
%p A327847      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k, p+j)*
%p A327847       binomial(binomial(k+i-1, i), j), j=0..n/i)))
%p A327847     end:
%p A327847 a:= n-> (k-> add(b(n$2, i, 0)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
%p A327847 seq(a(n), n=0..37);
%Y A327847 Column k=2 of A327245.
%K A327847 nonn
%O A327847 0,3
%A A327847 _Alois P. Heinz_, Sep 27 2019