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A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

%I A327674 #15 Aug 01 2021 06:45:12
%S A327674 1,1,3,19,121,1041,11191,130663,1731969,25778161,432791371,7752723771,
%T A327674 151553121193,3178030999729,71244609480591,1716351868658911,
%U A327674 43661944977384961,1173984102030774753,33302371396771085779,991402105480284394531,30912472614894951462681
%N A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).
%C A327674 Differs from A293840 and from A294253 first at n = 6.
%H A327674 Alois P. Heinz, <a href="/A327674/b327674.txt">Table of n, a(n) for n = 0..200</a>
%F A327674 a(n) = A327673(n,n).
%e A327674 a(3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.
%p A327674 b:= proc(n, i, k, p) option remember;
%p A327674      `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
%p A327674       b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
%p A327674     end:
%p A327674 a:= n-> add(b(n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):
%p A327674 seq(a(n), n=0..21);
%t A327674 b[n_, i_, k_, p_] := b[n, i, k, p] =
%t A327674      If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*
%t A327674      b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
%t A327674 a[n_] := Sum[b[n, n, i, 0] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
%t A327674 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)
%Y A327674 Main diagonal of A327673.
%Y A327674 Cf. A293840, A294253.
%K A327674 nonn
%O A327674 0,3
%A A327674 _Alois P. Heinz_, Sep 21 2019