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A325930 Total number of colors used in all colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and the colors span an initial interval of the color palette.

%I A325930 #12 Dec 15 2020 16:27:54
%S A325930 0,1,7,73,1075,21066,527122,16313963,609352653,26938878757,
%T A325930 1387465470527,82169954359252,5534425340505464,419977314311140561,
%U A325930 35617039966665620743,3352008343756176938273,347915661537105210844323,39607489635223003610928042
%N A325930 Total number of colors used in all colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and the colors span an initial interval of the color palette.
%H A325930 Alois P. Heinz, <a href="/A325930/b325930.txt">Table of n, a(n) for n = 0..296</a>
%F A325930 a(n) = Sum_{k=1..n} k * A322670(n,k).
%p A325930 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p A325930       binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
%p A325930     end:
%p A325930 a:= n-> add(k*add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p A325930 seq(a(n), n=0..18);
%t A325930 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k, j], {j, 1, Min[k, n]}]];
%t A325930 a[n_] := Sum[k Sum[b[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A325930 a /@ Range[0, 18] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A325930 Cf. A322670.
%K A325930 nonn
%O A325930 0,3
%A A325930 _Alois P. Heinz_, Sep 08 2019