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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. 4

%I #20 Aug 21 2021 06:35:00

%S 1,5,1,19,10,1,75,57,17,1,323,285,145,26,1,1512,1421,975,317,37,1,

%T 7630,7395,5999,2865,616,50,1,41245,40726,36183,22411,7315,1094,65,1,

%U 237573,237759,221689,163488,72581,16630,1812,82,1,1451359,1468162,1405001,1160764,649723,206249,34425,2840,101,1

%N 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 Alois P. Heinz, <a href="/A286232/b286232.txt">Rows n = 1..100, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e 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 Triangle T(n,k) begins:

%e 1;

%e 5, 1;

%e 19, 10, 1;

%e 75, 57, 17, 1;

%e 323, 285, 145, 26, 1;

%e 1512, 1421, 975, 317, 37, 1;

%e 7630, 7395, 5999, 2865, 616, 50, 1;

%e 41245, 40726, 36183, 22411, 7315, 1094, 65, 1;

%e ...

%t 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 setPartitions[n_] := setPartitions[n] = If[n == 1, {{{1}}}, Flatten[app[#, n]& /@ setPartitions[n-1], 1]];

%t T[n_, k_] := Select[setPartitions[n], Length[#] >= k&][[All, -k]] // Flatten // Total;

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 21 2021 *)

%Y Column k=1 gives A285424.

%Y Main diagonal and first lower diagonal give: A000012, A002522.

%Y Row sums give A000110(n) * A000217(n) = A105488(n+3).

%Y Cf. A285362, A286231, A286416.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, May 04 2017

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