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A285362
Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
18
1, 4, 2, 15, 12, 3, 60, 58, 28, 4, 262, 273, 185, 55, 5, 1243, 1329, 1094, 495, 96, 6, 6358, 6839, 6293, 3757, 1148, 154, 7, 34835, 37423, 36619, 26421, 11122, 2380, 232, 8, 203307, 217606, 219931, 180482, 96454, 28975, 4518, 333, 9, 1257913, 1340597, 1376929, 1230737, 787959, 308127, 67898, 7995, 460, 10
OFFSET
1,2
LINKS
EXAMPLE
T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
Triangle T(n,k) begins:
1;
4, 2;
15, 12, 3;
60, 58, 28, 4;
262, 273, 185, 55, 5;
1243, 1329, 1094, 495, 96, 6;
6358, 6839, 6293, 3757, 1148, 154, 7;
34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
...
MAPLE
T:= proc(h) option remember; local b; b:=
proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
+[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
end:
seq(T(n), n=1..12);
MATHEMATICA
T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
CROSSREFS
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
T(2n+1,n+1) gives A285410.
Sequence in context: A299789 A121662 A130042 * A109922 A090640 A302213
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 17 2017
STATUS
approved