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A285362
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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.
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18
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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]]]];
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CROSSREFS
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Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
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KEYWORD
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AUTHOR
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STATUS
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approved
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