A120058 -id:A120058 - cp's OEIS frontend

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

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: 0

Philippe Deléham, Feb 27 2012

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

			Triangle begins :
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
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Cf. Columns: A000027, A000290, A006331, A112742.

Cf. Diagonals: A011782, 2*A045623,

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

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).

A120095 Triangle T(n,k) = total of number at last index for all set partitions of n into k parts.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 11, 15, 4, 1, 23, 57, 34, 5, 1, 47, 195, 200, 65, 6, 1, 95, 633, 1010, 550, 111, 7, 1, 191, 1995, 4704, 3850, 1281, 175, 8, 1, 383, 6177, 20874, 24255, 11886, 2646, 260, 9, 1, 767, 18915, 89800, 143115, 97272, 31458, 4992, 369, 10

Offset: 1

Franklin T. Adams-Watters, Jun 07 2006

			The set partitions of 4 objects into 2 parts are {1,1,1,2}, {1,1,2,1}, {1,1,2,2}, {1,2,1,1}, {1,2,1,2}, {1,2,2,1} and {1,2,2,2}. The last terms of these sum to 2+1+2+1+2+1+2 = 11, so T(4,2) = 11.
Table starts:
  1;
  1,  2;
  1,  5,   3;
  1, 11,  15,   4;
  1, 23,  57,  34,  5;
  1, 47, 195, 200, 65, 6;
  ...

Alois P. Heinz, Rows n = 1..150, flattened

Cf. A008277, A120058.

Row sums are A087648(n-1).

A120095:= func< n,k | (&+[Binomial(j+k,j+1)*StirlingSecond(n-1,k+j-1): j in [0..1]]) >;
[A120095(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, May 03 2023

b:= proc(n, m) option remember; `if`(n=0, 1, add((t->
     `if`(n=1, j*x^t, b(n-1, t)))(max(m, j)), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 02 2021

b[n_, m_]:= b[n, m]= If[n==0, 1, Sum[
     If[n==1, j*x^#, b[n-1, #]]&[Max[m, j]], {j,m+1}]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, n}]&[b[n, 0]];
Table[T[n], {n,10}]//Flatten (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

def A120095(n,k):
    return sum(binomial(j+k,j+1)*stirling_number2(n-1,k+j-1) for j in range(2))
flatten([[A120095(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, May 03 2023

T(n,k) = (k*(k+1)/2)*S2(n-1,k) + k*S2(n-1,k-1) = 1/2 (S2(n+1,k) + S2(n,k) - S2(n-1,k-2)) = k T(n-1,k) + T(n-1,k-1) + S2(n-2,k-2), where S2 is the Stirling numbers of the second kind (A008277).

cp's OEIS Frontend

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

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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

Mathematica

Formula

A120095 Triangle T(n,k) = total of number at last index for all set partitions of n into k parts.

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Programs

Magma

Maple

Mathematica

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