%I #15 Aug 01 2021 06:45:12

%S 1,1,3,19,121,1041,11191,130663,1731969,25778161,432791371,7752723771,

%T 151553121193,3178030999729,71244609480591,1716351868658911,

%U 43661944977384961,1173984102030774753,33302371396771085779,991402105480284394531,30912472614894951462681

%N 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 Differs from A293840 and from A294253 first at n = 6.

%H Alois P. Heinz, <a href="/A327674/b327674.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A327673(n,n).

%e a(3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.

%p b:= proc(n, i, k, p) option remember;

%p `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*

%p b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))

%p end:

%p a:= n-> add(b(n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):

%p seq(a(n), n=0..21);

%t b[n_, i_, k_, p_] := b[n, i, k, p] =

%t If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*

%t b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];

%t a[n_] := Sum[b[n, n, i, 0] (-1)^(n-i) Binomial[n, i], {i, 0, n}];

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)

%Y Main diagonal of A327673.

%Y Cf. A293840, A294253.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 21 2019