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A327675 Number of 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).

%I A327675 #10 Apr 11 2022 08:36:10
%S A327675 1,1,4,40,355,4425,69521,1162951,22259414,478096938,11614132907,
%T A327675 299700810545,8456607358157,255883964141333,8275199908539114,
%U A327675 287869753459458468,10564476589147507523,409845503129745719513,16777378294629533764699,720626728499888159724831
%N A327675 Number of 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 A327675 Alois P. Heinz, <a href="/A327675/b327675.txt">Table of n, a(n) for n = 0..200</a>
%p A327675 b:= proc(n, i, k, p) option remember;
%p A327675      `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
%p A327675       b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
%p A327675     end:
%p A327675 a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*
%p A327675         binomial(k, i), i=0..k), k=0..n):
%p A327675 seq(a(n), n=0..23);
%t A327675 b[n_, i_, k_, p_] := b[n, i, k, p] =
%t A327675      If[n == 0, p!, If[i < 1, 0, Sum[ Binomial[k^i, j]*
%t A327675      b[n - i*j, Min[n - i*j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
%t A327675 a[n_] := Sum[Sum[b[n, n, i, 0]*(-1)^(k - i)*
%t A327675      Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A327675 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 11 2022, after _Alois P. Heinz_ *)
%Y A327675 Row sums of A327673.
%K A327675 nonn
%O A327675 0,3
%A A327675 _Alois P. Heinz_, Sep 21 2019