A038561 -id:A038561 - cp's OEIS frontend

A286416 Number T(n,k) of entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 8, 6, 1, 24, 25, 10, 1, 83, 98, 63, 15, 1, 324, 399, 338, 135, 21, 1, 1400, 1746, 1727, 980, 257, 28, 1, 6609, 8271, 8874, 6426, 2455, 448, 36, 1, 33758, 42284, 47191, 40334, 20506, 5474, 730, 45, 1, 185136, 231939, 263458, 250839, 158827, 57239, 11128, 1128, 55, 1

Offset: 1

Alois P. Heinz, May 08 2017

			T(3,2) = 6 because the number of entries in the second last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+2+2+1+1 = 6.
Triangle T(n,k) begins:
     1;
     3,    1;
     8,    6,    1;
    24,   25,   10,    1;
    83,   98,   63,   15,    1;
   324,  399,  338,  135,   21,   1;
  1400, 1746, 1727,  980,  257,  28,  1;
  6609, 8271, 8874, 6426, 2455, 448, 36, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Columns k=1-2 give: A038561 (for n>1), A286433.
Main diagonal and first lower diagonal give: A000012, A000217.
Row sums give A070071.
Cf. A000110, A130534, A270236, A286232.

A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 5, 6, 1, 4, 4, 8, 15, 21, 1, 5, 5, 11, 24, 52, 82, 1, 6, 6, 14, 33, 83, 203, 354, 1, 7, 7, 17, 42, 114, 324, 877, 1671, 1, 8, 8, 20, 51, 145, 445, 1400, 4140, 8536, 1, 9, 9, 23, 60, 176, 566, 1923, 6609, 21147, 46814

Offset: 0

Peter Luschny, Mar 28 2022

The array defines a family of Bell-like sequences. The case n = 1 are the Bell numbers A000110, case n = 0 is A032347 and case n = 2 is A038561. The n-th sequence r(k) = T(n, k) is defined for k >= 0 by the recurrence r(k) = Sum_{j=0..k-1} binomial(k-1, j)*r(j) with r(0) = 1 and r(1) = n.

			Array starts:
n\k 0, 1,  2,  3,  4,   5,    6,    7,     8,      9, ...
---------------------------------------------------------
[0] 1, 0,  1,  2,  6,  21,   82,  354,  1671,   8536, ... A032347
[1] 1, 1,  2,  5, 15,  52,  203,  877,  4140,  21147, ... A000110
[2] 1, 2,  3,  8, 24,  83,  324, 1400,  6609,  33758, ... A038561
[3] 1, 3,  4, 11, 33, 114,  445, 1923,  9078,  46369, ... A038559
[4] 1, 4,  5, 14, 42, 145,  566, 2446, 11547,  58980, ... A352683
[5] 1, 5,  6, 17, 51, 176,  687, 2969, 14016,  71591, ...
[6] 1, 6,  7, 20, 60, 207,  808, 3492, 16485,  84202, ...
[7] 1, 7,  8, 23, 69, 238,  929, 4015, 18954,  96813, ...
[8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423, 109424, ...
[9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, ...

Rows: A032347, A000110 (Bell), A038561, A038559, A352683.
Diagonals: A352684 (main).
Cf. A040027 (Gould), A352686 (subtriangle).
Compare A352680 for a similar array based on the Catalan numbers.

function BellRow(m, len)
    a = m; P = BigInt[1]; T = BigInt[1]
    for n in 1:len
        T = vcat(T, a)
        P = cumsum(vcat(a, P))
        a = P[end]
    end
T end
for n in 0:9 BellRow(n, 9) |> println end

alias(PS = ListTools:-PartialSums):
BellRow := proc(n, len) local a, k, P, T;
a := n; P := [1]; T := [1];
for k from 1 to len-1 do
   T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
T end: seq(lprint(BellRow(n, 10)), n = 0..9);

nmax = 10;
BellRow[n_, len_] := Module[{a, k, P, T}, a = n; P = {1}; T = {1};
   For[k = 1, k <= len - 1, k++,
      T = Append[T, a]; P = Accumulate[Join[{a}, P]]; a = P[[-1]]];
   T];
rows = Table[BellRow[n, nmax + 1], {n, 0, nmax}];
A[n_, k_] := rows[[n + 1, k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array with length k can be computed by the following procedure:
     A = [n], P = [1], R = [1];
     Repeat k-1 times: R = [R, A], P = PS([A, P]), A = [P[end]];
     Return R.

A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

Original entry on oeis.org

1, 2, 3, 3, 5, 8, 8, 11, 16, 24, 24, 32, 43, 59, 83, 83, 107, 139, 182, 241, 324, 324, 407, 514, 653, 835, 1076, 1400, 1400, 1724, 2131, 2645, 3298, 4133, 5209, 6609, 6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758

Offset: 0

N. J. A. Sloane, R. K. Guy

			Triangle starts:
[0] [   1]
[1] [   2,    3]
[2] [   3,    5,    8]
[3] [   8,   11,   16,    24]
[4] [  24,   32,   43,    59,    83]
[5] [  83,  107,  139,   182,   241,   324]
[6] [ 324,  407,  514,   653,   835,  1076,  1400]
[7] [1400, 1724, 2131,  2645,  3298,  4133,  5209,  6609]
[8] [6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758]

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Borders give A038561.

Cf. A011971.

a046937 n k = a046937_tabl !! n !! k
a046937_row n = a046937_tabl !! n
a046937_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [2,3]
-- Reinhard Zumkeller, Jan 13 2013

# Compare the analogue algorithm for the Catalan triangle in A350584.
A046937Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := P; T := [op(T), P] od; T end:
A046937Triangle(9): ListTools:-Flatten(%); # Peter Luschny, Mar 27 2022

a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 06 2013 *)

A352685 Array of Aitken-Bell triangles of order m (read by rows) read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 4, 4, 3, 3, 2, 1, 5, 5, 4, 5, 5, 2, 1, 6, 6, 5, 7, 8, 5, 3, 1, 7, 7, 6, 9, 11, 8, 7, 4, 1, 8, 8, 7, 11, 14, 11, 11, 10, 6, 1, 9, 9, 8, 13, 17, 14, 15, 16, 15, 6, 1, 10, 10, 9, 15, 20, 17, 19, 22, 24, 15, 8, 1, 11, 11, 10, 17, 23, 20, 23, 28, 33, 24, 20, 11, 1, 12, 12, 11, 19, 26, 23, 27, 34, 42, 33, 32, 27, 15

Offset: 0

Peter Luschny, Mar 29 2022

An Aitken-Bell triangle of order m is defined by T(0, 0) = 1, T(1, 0) = m, T(n, 0) = T(n-1, n-1) and T(n, k) = T(n, k-1) + T(n-1, k-1), for n >= 0 and 0 <= k <= n. The case m = 1 is Aitken's array A011971 with the first column the Bell numbers A000110, case m = 0 is the triangle A046934 with the first column A032347 and case m = 2 is the triangle A046937 with the first column A038561.

			Array starts:
[0] 1, 0,  1,  1,  1,  2,  2,  3,  4,  6,  6,   8,  11,  15, ... A046934
[1] 1, 1,  2,  2,  3,  5,  5,  7, 10, 15, 15,  20,  27,  37, ... A011971
[2] 1, 2,  3,  3,  5,  8,  8, 11, 16, 24, 24,  32,  43,  59, ... A046937
[3] 1, 3,  4,  4,  7, 11, 11, 15, 22, 33, 33,  44,  59,  81, ...
[4] 1, 4,  5,  5,  9, 14, 14, 19, 28, 42, 42,  56,  75, 103, ...
[5] 1, 5,  6,  6, 11, 17, 17, 23, 34, 51, 51,  68,  91, 125, ...
[6] 1, 6,  7,  7, 13, 20, 20, 27, 40, 60, 60,  80, 107, 147, ...
[7] 1, 7,  8,  8, 15, 23, 23, 31, 46, 69, 69,  92, 123, 169, ...
[8] 1, 8,  9,  9, 17, 26, 26, 35, 52, 78, 78, 104, 139, 191, ...
[9] 1, 9, 10, 10, 19, 29, 29, 39, 58, 87, 87, 116, 155, 213, ...

The main diagonals of the triangles are in A352682.

Cf. A000110, A046934, A011971, A046937, A032347, A038561.

function BellTriangle(m, len)
    a = m; P = [1]; T = []
    for n in 1:len
        T = vcat(T, P)
        P = cumsum(vcat(a, P))
        a = P[end]
    end
T end
for n in 0:9 BellTriangle(n, 4) |> println end

alias(PS = ListTools:-PartialSums):
BellTriangle := proc(m, len) local a, k, P, T; a := m; P := [1]; T := [];
for n from 1 to len  do T := [op(T), P]; P := PS([a, op(P)]); a := P[-1] od;
ListTools:-Flatten(T) end:
for n from 0 to 9 do print(BellTriangle(n, 5)) od; # Prints array by rows.

nmax = 13;
row[m_] := row[m] = Module[{T}, T[0, 0] = 1; T[1, 0] = m; T[n_, 0] := T[n, 0] = T[n-1, n-1]; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k-1]; Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten];
A[n_, k_] := row[n][[k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2024 *)

Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. The Aitken-Bell triangle T of order m with n rows can be computed by the following procedure:
     A = [m], P = [1], T = [];
     Repeat n times: T = [T, P], P = PS([A, P]), A = [P[end]];
     Return T.

cp's OEIS Frontend

A286416 Number T(n,k) of entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

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A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

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A352685 Array of Aitken-Bell triangles of order m (read by rows) read by ascending antidiagonals.

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