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

Author

Peter Luschny, Mar 29 2022

Keywords

• nonn
• tabl

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.

Examples

			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, ...
		

Crossrefs

The main diagonals of the triangles are in A352682.

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

Programs

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

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 *)

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.