A046937 -id:A046937 - cp's OEIS frontend

A038561 Left-hand border of triangle A046937.

1, 2, 3, 8, 24, 83, 324, 1400, 6609, 33758, 185136, 1083233, 6726366, 44130128, 304741623, 2207682188, 16729947276, 132281116715, 1088831511000, 9311082630620, 82569723552561, 758057178490082, 7194283782101844, 70481938088367569

Offset: 0

Henry Gould, N. J. A. Sloane

For n>1: a(n) is the number of entries in the last blocks of all set partitions of [n]. a(3) = 8 because the number of entries in the last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+1+1+2+1 = 8. - Alois P. Heinz, May 08 2017

H. W. Gould, A linear binomial recurrence and the Bell numbers and polynomials, preprint, 1998

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968

A040027(n) + B(n), where B(n) = Bell numbers A000110.
Related to A000110, A040027, A038559, A038560.
Column k=1 of A286416 (for n>1).

a038561 = head . a046937_row  -- Reinhard Zumkeller, Jan 06 2014

A038561List := proc(m) local A, P, n; A := [1,2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A038561List(24); # Peter Luschny, Mar 24 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]; a[n_] := a[n, 0]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 06 2013 *)

G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jun 30 2020

A046938 Sequence formed from rows of triangle A046937.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 16, 24, 32, 43, 59, 83, 107, 139, 182, 241, 324, 407, 514, 653, 835, 1076, 1400, 1724, 2131, 2645, 3298, 4133, 5209, 6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758, 40367, 48376, 58109, 69973, 84482, 102289

Offset: 0

N. J. A. Sloane

nonn

Apparently formed by deleting any repeated elements in A046937. - Sean A. Irvine, May 05 2021

Title corrected by Sean A. Irvine, May 05 2021

A350584 Triangle read by rows, T(n, k) = [x^k] ((2x^3 - 3x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 5, 12, 19, 1, 6, 18, 37, 56, 1, 7, 25, 62, 118, 174, 1, 8, 33, 95, 213, 387, 561, 1, 9, 42, 137, 350, 737, 1298, 1859, 1, 10, 52, 189, 539, 1276, 2574, 4433, 6292, 1, 11, 63, 252, 791, 2067, 4641, 9074, 15366, 21658

Offset: 1

Peter Luschny, Mar 27 2022

			Triangle starts:
[1] [1]
[2] [1,  3]
[3] [1,  4,  7]
[4] [1,  5, 12,  19]
[5] [1,  6, 18,  37,  56]
[6] [1,  7, 25,  62, 118,  174]
[7] [1,  8, 33,  95, 213,  387,  561]
[8] [1,  9, 42, 137, 350,  737, 1298, 1859]
[9] [1, 10, 52, 189, 539, 1276, 2574, 4433, 6292]

A280891 (row sums), A135339 (alternating row sums), A005807 or A071716 (main diagonal).

# Compare the analogue algorithm for the Bell triangle in A046937.
A350584Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), A[-1]]);
A := P; T := [op(T), P] od; T end:
A350584Triangle(10): ListTools:-Flatten(%);
# Alternative:
ogf := n -> (2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2):
ser := n -> series(ogf(n), x, n):
row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
seq(row(n), n = 1..10);

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

A038561 Left-hand border of triangle A046937.

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A046938 Sequence formed from rows of triangle A046937.

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A350584 Triangle read by rows, T(n, k) = [x^k] ((2x^3 - 3x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

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Maple

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

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Julia

Maple

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cp's OEIS Frontend

A038561 Left-hand border of triangle A046937.

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References

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Haskell

Maple

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A046938 Sequence formed from rows of triangle A046937.

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Author

Keywords

Comments

Extensions

A350584 Triangle read by rows, T(n, k) = [x^k] ((2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Julia

Maple

Mathematica

Formula

A350584 Triangle read by rows, T(n, k) = [x^k] ((2x^3 - 3x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.