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A286232 Sum T(n,k) of the entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A286232 #20 Aug 21 2021 06:35:00
%S A286232 1,5,1,19,10,1,75,57,17,1,323,285,145,26,1,1512,1421,975,317,37,1,
%T A286232 7630,7395,5999,2865,616,50,1,41245,40726,36183,22411,7315,1094,65,1,
%U A286232 237573,237759,221689,163488,72581,16630,1812,82,1,1451359,1468162,1405001,1160764,649723,206249,34425,2840,101,1
%N A286232 Sum T(n,k) of the entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A286232 Alois P. Heinz, <a href="/A286232/b286232.txt">Rows n = 1..100, flattened</a>
%H A286232 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A286232 T(3,2) = 10 because the sum of the entries in the second last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+4+1+2 = 10.
%e A286232 Triangle T(n,k) begins:
%e A286232       1;
%e A286232       5,     1;
%e A286232      19,    10,     1;
%e A286232      75,    57,    17,     1;
%e A286232     323,   285,   145,    26,    1;
%e A286232    1512,  1421,   975,   317,   37,    1;
%e A286232    7630,  7395,  5999,  2865,  616,   50,  1;
%e A286232   41245, 40726, 36183, 22411, 7315, 1094, 65, 1;
%e A286232   ...
%t A286232 app[P_, n_] := Module[{P0}, Table[P0 = Append[P, {}]; AppendTo[P0[[i]], n]; If[Last[P0] == {}, Most[P0], P0], {i, 1, Length[P]+1}]];
%t A286232 setPartitions[n_] := setPartitions[n] = If[n == 1, {{{1}}}, Flatten[app[#, n]& /@ setPartitions[n-1], 1]];
%t A286232 T[n_, k_] := Select[setPartitions[n], Length[#] >= k&][[All, -k]] // Flatten // Total;
%t A286232 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 21 2021 *)
%Y A286232 Column k=1 gives A285424.
%Y A286232 Main diagonal and first lower diagonal give: A000012, A002522.
%Y A286232 Row sums give A000110(n) * A000217(n) = A105488(n+3).
%Y A286232 Cf. A285362, A286231, A286416.
%K A286232 nonn,tabl
%O A286232 1,2
%A A286232 _Alois P. Heinz_, May 04 2017