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A327677 Number of colored compositions of n using all colors of an n-set such that any part i has a color pattern of i (distinct) colors in increasing order.

%I A327677 #12 Aug 01 2021 06:45:15
%S A327677 1,1,3,13,71,481,3861,35743,373591,4347103,55671713,777540523,
%T A327677 11754153869,191114449579,3324296885339,61575268263193,
%U A327677 1209681079172663,25116819005925409,549458325556099551,12629191765880480035,304232436498153748441,7663883684722855430077
%N A327677 Number of colored compositions of n using all colors of an n-set such that any part i has a color pattern of i (distinct) colors in increasing order.
%H A327677 Alois P. Heinz, <a href="/A327677/b327677.txt">Table of n, a(n) for n = 0..444</a>
%e A327677 a(3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
%p A327677 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A327677       add(b(n-i*j, i-1, p+j)*binomial(n, i*j), j=0..n/i)))
%p A327677     end:
%p A327677 a:= n-> b(n$2, 0):
%p A327677 seq(a(n), n=0..23);
%t A327677 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
%t A327677      Sum[b[n - i*j, i - 1, p + j]*Binomial[n, i*j], {j, 0, n/i}]]];
%t A327677 a[n_] := b[n, n, 0];
%t A327677 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)
%Y A327677 Cf. A261774.
%K A327677 nonn
%O A327677 0,3
%A A327677 _Alois P. Heinz_, Sep 21 2019