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%I #11 Apr 11 2022 09:13:09
%S 0,1,7,96,1120,17133,318550,6174409,134222746,3227634290,86758393637,
%T 2455321082763,75378081090618,2465367832093301,85698931156186485,
%U 3188667759545387936,124641366048950285559,5131034234304472735967,222142129822095164108159
%N Total number of colors in all colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order).
%H Alois P. Heinz, <a href="/A327676/b327676.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=1..n} k * A327673(n,k).
%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(add(k*b(n$2, i, 0)*(-1)^(k-i)*
%p binomial(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..23);
%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[Sum[k*b[n, n, i, 0]*(-1)^(k-i)*
%t Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 11 2022, after _Alois P. Heinz_ *)
%Y Cf. A327673.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 21 2019
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