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A327588 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 have i colors in (weakly) increasing order.

%I A327588 #11 Apr 11 2022 08:14:29
%S A327588 0,1,7,62,642,7784,108824,1725072,30605384,601213744,12958778704,
%T A327588 304145108160,7722286425312,210920029636224,6166996162239840,
%U A327588 192199468584942816,6360760834966301120,222782888877269937664,8233066075880951824000,320162458265691237967360
%N A327588 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 have i colors in (weakly) increasing order.
%H A327588 Alois P. Heinz, <a href="/A327588/b327588.txt">Table of n, a(n) for n = 0..200</a>
%F A327588 a(n) = Sum_{k=1..n} k * A327245(n,k).
%p A327588 C:= binomial:
%p A327588 b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p A327588       b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
%p A327588     end:
%p A327588 a:= n-> add(k*add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
%p A327588 seq(a(n), n=0..21);
%t A327588 c = Binomial;
%t A327588 b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
%t A327588      b[n - i*j, Min[n-i*j, i-1], k, p+j]*c[c[k+i-1, i], j], {j, 0, n/i}]]];
%t A327588 a[n_] := Sum[k*Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
%t A327588 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Apr 11 2022, after _Alois P. Heinz_ *)
%Y A327588 Cf. A327245.
%K A327588 nonn
%O A327588 0,3
%A A327588 _Alois P. Heinz_, Sep 17 2019