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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
1, 5, 1, 19, 10, 1, 75, 57, 17, 1, 323, 285, 145, 26, 1, 1512, 1421, 975, 317, 37, 1, 7630, 7395, 5999, 2865, 616, 50, 1, 41245, 40726, 36183, 22411, 7315, 1094, 65, 1, 237573, 237759, 221689, 163488, 72581, 16630, 1812, 82, 1, 1451359, 1468162, 1405001, 1160764, 649723, 206249, 34425, 2840, 101, 1
OFFSET
1,2
LINKS
EXAMPLE
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.
Triangle T(n,k) begins:
1;
5, 1;
19, 10, 1;
75, 57, 17, 1;
323, 285, 145, 26, 1;
1512, 1421, 975, 317, 37, 1;
7630, 7395, 5999, 2865, 616, 50, 1;
41245, 40726, 36183, 22411, 7315, 1094, 65, 1;
...
MATHEMATICA
app[P_, n_] := Module[{P0}, Table[P0 = Append[P, {}]; AppendTo[P0[[i]], n]; If[Last[P0] == {}, Most[P0], P0], {i, 1, Length[P]+1}]];
setPartitions[n_] := setPartitions[n] = If[n == 1, {{{1}}}, Flatten[app[#, n]& /@ setPartitions[n-1], 1]];
T[n_, k_] := Select[setPartitions[n], Length[#] >= k&][[All, -k]] // Flatten // Total;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 21 2021 *)
CROSSREFS
Column k=1 gives A285424.
Main diagonal and first lower diagonal give: A000012, A002522.
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Sequence in context: A349544 A193857 A146055 * A147437 A147369 A375363
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 04 2017
STATUS
approved