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A285365 Sum of the entries in the third blocks of all set partitions of [n].

%I A285365 #9 May 27 2018 10:32:22
%S A285365 3,28,185,1094,6293,36619,219931,1376929,9023266,61944014,445076570,
%T A285365 3341575188,26164558199,213243368898,1805626838935,15856747810014,
%U A285365 144189514375955,1355629263039685,13159535002316403,131729480987412527,1358188539892586220
%N A285365 Sum of the entries in the third blocks of all set partitions of [n].
%H A285365 Alois P. Heinz, <a href="/A285365/b285365.txt">Table of n, a(n) for n = 3..400</a>
%H A285365 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A285365 a(n) = A285362(n,3).
%e A285365 a(3) = 3 because the sum of the entries in the third blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+0+0+0+3 = 3.
%p A285365 a:= proc(h) option remember; local b; b:=
%p A285365       proc(n, m) option remember;
%p A285365         `if`(n=0, [1, 0], add((p-> `if`(j=3, p+ [0,
%p A285365         (h-n+1)*p[1]], p))(b(n-1, max(m, j))), j=1..m+1))
%p A285365       end: b(h, 0)[2]
%p A285365     end:
%p A285365 seq(a(n), n=3..30);
%t A285365 a[h_] := a[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, If[j == 3, p + {0, (h - n + 1)*p[[1]]}, p]][b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; b[h, 0][[2]]];
%t A285365 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A285365 Column k=3 of A285362.
%K A285365 nonn
%O A285365 3,1
%A A285365 _Alois P. Heinz_, Apr 17 2017