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

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

0,8

Comments

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.

Links

Table of n, a(n) for n=0..104.

Formula

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.

Example

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, ...

Maple

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.

Mathematica

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 *)

Prog

(Julia)
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

Crossrefs

The main diagonals of the triangles are in A352682.

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

Sequence in context: A181935 A027358 A332548 * A191781 A155092 A095133

Adjacent sequences: A352682 A352683 A352684 * A352686 A352687 A352688

Keyword

nonn,tabl

Author

Peter Luschny, Mar 29 2022

Status

approved