A120095 -id:A120095 - cp's OEIS frontend

A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

1, 3, 9, 31, 120, 514, 2407, 12205, 66491, 386699, 2388096, 15589732, 107165081, 773106715, 5836100685, 45981026703, 377230766908, 3215977070706, 28437411817135, 260380616093533, 2464930698184351, 24091925888687459, 242802079705721156, 2520198597834860148

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - Franklin T. Adams-Watters, Jun 07 2006

Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets. Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - Alois P. Heinz, Jul 30 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000110, A035098, A059606.
Main diagonal of A120057, row sums of A120095.
Column 1 of array in A322770.
Row n=2 of A346520.

[(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* Zerinvary Lajos, Mar 31 2007 *)
(#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0,30]],3,1] (* Harvey P. Dale, Jul 20 2021 *)

A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

Original entry on oeis.org

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

Offset: 1

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

			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;
  ...

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

Cf. A120058, A120095. First column is A000110.
Main diagonal is A087648(n-1).
Row sums give A346772.

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

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-François Alcover, Apr 08 2016, after Alois P. Heinz *)

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

A120058 Coefficients for obtaining A120057 from Bell numbers.

Original entry on oeis.org

1, 2, -1, 3, -4, 2, 4, -9, 10, -4, 5, -16, 28, -24, 8, 6, -25, 60, -80, 56, -16, 7, -36, 110, -200, 216, -128, 32, 8, -49, 182, -420, 616, -560, 288, -64, 9, -64, 280, -784, 1456, -1792, 1408, -640, 128, 10, -81, 408, -1344, 3024, -4704, 4992, -3456, 1408, -256

Offset: 1

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

Appears to be essentially the same as A056863, but (as of Jun 06 2006) that sequence definition is unclear and there are discrepencies in the signs.

Alternating column sums appear to be 3^n.

			Table starts:
1
2,-1
3,-4,2
4,-9,10,-4
5,-16,28,-24,8
6,-25,60,-80,56,-16

Cf A120057, A000110, A056863.
Cf. A008275, A120095.
Cf. A134315.

T[n_, 1] := n; T[n_, n_] := (-1)^(n+1)*2^(n-2); T[n_, k_] /; 2 <= k <= n-1 := T[n, k] = 2*T[n-1, k] - 2*T[n-1, k-1] + 2*T[n-2, k-1] - T[n-2, k]; T[, ] = 0; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2016, after Philippe Deléham *)

A120057(n,k) = sum_{i=1,k} T(n,i)*B(n-i+1).
T(n,k) = Sum_j A120095(n,j) * S1(j,n-k+1), where S1 is the Stirling numbers of the first kind (A008275).
Unsigned version, as an infinite lower triangular matrix, equals A007318 * A134315. - Gary W. Adamson, Oct 19 2007
T(n,k) = 2*T(n-1,k) - 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k). - Philippe Deléham, Feb 27 2012

cp's OEIS Frontend

A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

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Magma

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A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

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Maple

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A120058 Coefficients for obtaining A120057 from Bell numbers.

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