A120057 -id:A120057 - cp's OEIS frontend

A120058 Coefficients for obtaining A120057 from Bell numbers.

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

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

Original entry on oeis.org

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 *)

A346772 Total sum of block indices of the elements over all partitions of [n].

Original entry on oeis.org

0, 1, 5, 22, 100, 482, 2475, 13527, 78476, 481687, 3117962, 21218851, 151387882, 1129430737, 8790433999, 71222812912, 599577147056, 5235054113412, 47331036294905, 442462325254995, 4270909302907430, 42514043248222709, 435920900603529954, 4599155199953703373

Offset: 0

Alois P. Heinz, Aug 02 2021

nonn

			a(3) = 22 = 3 + 4 + 4 + 5 + 6, summing block indices 111, 112, 121, 122, 123 of the 5 partitions of [3]: 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..574
Wikipedia, Partition of a set

Row sums of A120057.

Cf. A000110, A070071, A105488, A126347, A270236.

b:= proc(n, m) option remember; `if`(n=0, [1, 0], add(
     (p-> p+[0, p[1]*j])(b(n-1, max(m, j))), j=1..m+1))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[
     Function[p, p+{0, p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} A120057(n,k).
a(n) = Sum_{k=0..n*(n-1)/2} (n+k) * A126347(n,k).
a(n) = Sum_{k=1..n} k * A270236(n,k).

cp's OEIS Frontend

A120058 Coefficients for obtaining A120057 from Bell numbers.

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

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A346772 Total sum of block indices of the elements over all partitions of [n].

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