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.

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

Author

Peter Luschny, Mar 28 2022

Keywords

• nonn
• tabl

Comments

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.

Examples

			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, ...
		

Crossrefs

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.

Programs

Julia

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

Maple

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);

Mathematica

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 *)

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. 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.