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A270495 Sum of the sizes of the third blocks in all set partitions of {1,2,...,n}.

%I A270495 #11 May 27 2018 02:00:32
%S A270495 1,8,47,253,1345,7304,41193,243152,1506521,9799547,66844755,477297022,
%T A270495 3560469469,27692022408,224128400923,1884299045789,16427961558365,
%U A270495 148293477761232,1384008870213057,13336887952918752,132535336519342301,1356662080571809755
%N A270495 Sum of the sizes of the third blocks in all set partitions of {1,2,...,n}.
%H A270495 Alois P. Heinz, <a href="/A270495/b270495.txt">Table of n, a(n) for n = 3..575</a>
%H A270495 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A270495 b:= proc(n, m) option remember; `if`(n=0, [1, 0],
%p A270495       add((p->`if`(j<4, [p[1], p[2]+p[1]*x^j], p))(
%p A270495        b(n-1, max(m, j))), j=1..m+1))
%p A270495     end:
%p A270495 a:= n-> coeff(b(n, 0)[2], x, 3):
%p A270495 seq(a(n), n=3..25);
%t A270495 b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, If[j < 4, {p[[1]], p[[2]] + p[[1]]*x^j}, p]][b[n - 1, Max[m, j]]], {j, 1, m + 1}]];
%t A270495 a[n_] := Coefficient[b[n, 0][[2]], x, 3];
%t A270495 Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)
%Y A270495 Column p=3 of A270236.
%K A270495 nonn
%O A270495 3,2
%A A270495 _Alois P. Heinz_, Mar 18 2016