%I #13 Dec 15 2020 16:28:24
%S 0,1,6,34,191,1208,7840,54152,377396,2868528,22719712,187318016,
%T 1594593876,13795808224,125535871760,1192418406800,11747646588912,
%U 118703814213296,1223646182128656,12755728151091424,137199027931128992,1527404635450188128,17599899510211606336
%N Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.
%H Alois P. Heinz, <a href="/A326656/b326656.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{k=1..n} k * A326500(n,k).
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
%p end:
%p a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..25);
%t b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k + t - 1, t]], {j, 0, n/i}]]];
%t a[n_] := Sum[k Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y Cf. A326500.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 12 2019