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A327680 Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

%I A327680 #12 Dec 18 2020 04:01:34
%S A327680 0,1,7,44,358,2904,29112,296448,3520568,43482208,602603120,8712724080,
%T A327680 138736978208,2302036052128,41417364992160,776413790063328,
%U A327680 15597709327298944,325945020056535968,7238587734613470208,166897326948551436384,4061690336695535982048
%N A327680 Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.
%H A327680 Alois P. Heinz, <a href="/A327680/b327680.txt">Table of n, a(n) for n = 0..200</a>
%F A327680 a(n) = Sum_{k=1..n} k * A309973(n,k).
%p A327680 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A327680       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
%p A327680        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
%p A327680     end:
%p A327680 a:= n-> add(add(k*b(n$2, i)*(-1)^(k-i)*
%p A327680         binomial(k, i), i=0..k), k=0..n):
%p A327680 seq(a(n), n=0..22);
%t A327680 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-1, i], j] j!, {j, 0, n/i}]]];
%t A327680 a[n_] := Sum[Sum[k b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A327680 a /@ Range[0, 22] (* _Jean-François Alcover_, Dec 18 2020, after_Alois P. Heinz_ *)
%Y A327680 Cf. A309973.
%K A327680 nonn
%O A327680 0,3
%A A327680 _Alois P. Heinz_, Sep 21 2019