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A120057 Table T(n,k) = sum over all set partitions of n of number at index k. 3
1, 2, 3, 5, 8, 9, 15, 25, 29, 31, 52, 89, 106, 115, 120, 203, 354, 431, 474, 499, 514, 877, 1551, 1923, 2141, 2273, 2355, 2407, 4140, 7403, 9318, 10489, 11224, 11695, 12002, 12205, 21147, 38154, 48635, 55286, 59595, 62434, 64331, 65614, 66491, 115975, 210803, 271617, 311469, 338019, 355951, 368205, 376665, 382559, 386699 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

T(n,k) = Sum_{i=1..k} A120058(n,i)*B(n-i+1), where B(n) are the Bell numbers, (A000110).

EXAMPLE

The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9.

Table starts:

1,

2,3,

5,8,9,

15,25,29,31,

52,89,106,115,120,

MAPLE

b:= proc(n, m) option remember; `if`(n=0, [1, 0],

      add((p-> [p[1], expand(p[2]*x+p[1]*j)])(

        b(n-1, max(m, j))), j=1..m+1))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]):

seq(T(n), n=1..10);  # Alois P. Heinz, Mar 24 2016

MATHEMATICA

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];

T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]];

Table[T[n], {n, 1, 10}] // Flatten (* Jean-Fran├žois Alcover, Apr 08 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A120058, A120095. First column is A000110.

Main diagonal is A087648(n-1).

Sequence in context: A295082 A168154 A278334 * A099422 A294913 A056903

Adjacent sequences:  A120054 A120055 A120056 * A120058 A120059 A120060

KEYWORD

nonn,tabl

AUTHOR

Franklin T. Adams-Watters, Jun 06 2006, Jun 07 2006

STATUS

approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)