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A350683 Total sum over all partitions of [n] of elements i contained in block i when blocks are ordered with decreasing largest elements.

%I A350683 #15 May 08 2022 08:44:55
%S A350683 0,1,1,8,17,98,362,1916,9512,53858,315872,1984979,13105685,91128546,
%T A350683 663815424,5055622309,40148341135,331753228115,2846786927873,
%U A350683 25323311882074,233137061978065,2218141402504254,21780561656373552,220451321425101091,2297330116404668422
%N A350683 Total sum over all partitions of [n] of elements i contained in block i when blocks are ordered with decreasing largest elements.
%H A350683 Alois P. Heinz, <a href="/A350683/b350683.txt">Table of n, a(n) for n = 0..575</a>
%H A350683 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A350683 a(n) = Sum_{k=1..max(0,A008805(n-1))} k * A350684(n,k).
%e A350683 a(4) = 17 = 3*1 + 4*2 + 2*3: 432(1), 42(1)|3, 4(1)|3|2, 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1, 43(1)|(2), 4(1)|3(2).
%p A350683 b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
%p A350683      `if`(n=j, p[1]*j, 0)]+p)(b(n-1, max(m, j))), j=1..m+1))
%p A350683     end:
%p A350683 a:= n-> b(n, 0)[2]:
%p A350683 seq(a(n), n=0..25);
%t A350683 b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[With[{p = b[n-1, Max[m, j]]},
%t A350683      {0, If[n == j, p[[1]]*j, 0]} + p], {j, 1, m+1}]];
%t A350683 a[n_] := b[n, 0][[2]];
%t A350683 Table[a[n], {n, 0, 25}]; (* _Jean-François Alcover_, May 08 2022, after _Alois P. Heinz_ *)
%Y A350683 Cf. A008805, A350648, A350684.
%K A350683 nonn
%O A350683 0,4
%A A350683 _Alois P. Heinz_, Jan 11 2022