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A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order.

%I A327118 #15 May 06 2020 05:27:52
%S A327118 0,1,5,24,129,752,4796,33117,246336,1961233,16626100,149376533,
%T A327118 1416602126,14130107135,147781380186,1616110614723,18434515499407,
%U A327118 218849548323400,2698686223271769,34504328470389166,456669361749612835,6247290917385938422,88216873775207493056
%N A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order.
%H A327118 Alois P. Heinz, <a href="/A327118/b327118.txt">Table of n, a(n) for n = 0..230</a>
%H A327118 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327118 a(n) = Sum_{k=1..n} k * A327117(n,k).
%p A327118 C:= binomial:
%p A327118 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327118       b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))
%p A327118     end:
%p A327118 a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
%p A327118 seq(a(n), n=0..25);
%t A327118 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k, i]+j-1, j], {j, 0, n/i}]]];
%t A327118 a[n_] := Sum[k Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A327118 a /@ Range[0, 25] (* _Jean-François Alcover_, May 06 2020, after Maple *)
%Y A327118 Cf. A327117.
%K A327118 nonn
%O A327118 0,3
%A A327118 _Alois P. Heinz_, Sep 13 2019