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A346772 Total sum of block indices of the elements over all partitions of [n].

%I A346772 #23 Apr 27 2022 08:47:35
%S A346772 0,1,5,22,100,482,2475,13527,78476,481687,3117962,21218851,151387882,
%T A346772 1129430737,8790433999,71222812912,599577147056,5235054113412,
%U A346772 47331036294905,442462325254995,4270909302907430,42514043248222709,435920900603529954,4599155199953703373
%N A346772 Total sum of block indices of the elements over all partitions of [n].
%H A346772 Alois P. Heinz, <a href="/A346772/b346772.txt">Table of n, a(n) for n = 0..574</a>
%H A346772 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A346772 a(n) = Sum_{k=1..n} A120057(n,k).
%F A346772 a(n) = Sum_{k=0..n*(n-1)/2} (n+k) * A126347(n,k).
%F A346772 a(n) = Sum_{k=1..n} k * A270236(n,k).
%e A346772 a(3) = 22 = 3 + 4 + 4 + 5 + 6, summing block indices 111, 112, 121, 122, 123 of the 5 partitions of [3]: 123, 12|3, 13|2, 1|23, 1|2|3.
%p A346772 b:= proc(n, m) option remember; `if`(n=0, [1, 0], add(
%p A346772      (p-> p+[0, p[1]*j])(b(n-1, max(m, j))), j=1..m+1))
%p A346772     end:
%p A346772 a:= n-> b(n, 0)[2]:
%p A346772 seq(a(n), n=0..25);
%t A346772 b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[
%t A346772      Function[p, p+{0, p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
%t A346772 a[n_] := b[n, 0][[2]];
%t A346772 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 27 2022, after _Alois P. Heinz_ *)
%Y A346772 Row sums of A120057.
%Y A346772 Cf. A000110, A070071, A105488, A126347, A270236.
%K A346772 nonn
%O A346772 0,3
%A A346772 _Alois P. Heinz_, Aug 02 2021