A375555 Triangle read by rows: T(n, k) = abs(A181937(k, n)), where A181937 are the André numbers, for 0 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 16, 9, 4, 1, 1, 1, 61, 19, 14, 5, 1, 1, 1, 272, 99, 34, 20, 6, 1, 1, 1, 1385, 477, 69, 55, 27, 7, 1, 1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1, 1, 1, 50521, 11259, 2896, 251, 209, 119, 44, 9, 1

Offset: 0

Author

Peter Luschny, Aug 19 2024

Keywords

• nonn
• tabl

Comments

See A181937 for comments and references.

Examples

			Triangle starts:
  [0]  1;
  [1]  1, 1;
  [2]  1, 1,    1;
  [3]  1, 1,    2,    1;
  [4]  1, 1,    5,    3,   1;
  [5]  1, 1,   16,    9,   4,   1;
  [6]  1, 1,   61,   19,  14,   5,  1;
  [7]  1, 1,  272,   99,  34,  20,  6,  1;
  [8]  1, 1, 1385,  477,  69,  55, 27,  7, 1;
  [9]  1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1;
.
Seen as an array:
  [0]  1, 1,      1,      1,      1,      1,      1,      1, ...
  [1]  1, 1,      2,      3,      4,      5,      6,      7, ...
  [2]  1, 1,      5,      9,     14,     20,     27,     35, ...
  [3]  1, 1,     16,     19,     34,     55,     83,    119, ...
  [4]  1, 1,     61,     99,     69,    125,    209,    329, ...
  [5]  1, 1,    272,    477,    496,    251,    461,    791, ...
  [6]  1, 1,   1385,   1513,   2896,   2300,    923,   1715, ...
  [7]  1, 1,   7936,  11259,  11056,  15775,  10284,   3431, ...
		

Crossrefs

Cf. A181937, A375554 (row sums), A030662 (central terms, main diagonal of array), A010763 (central terms of the (1, 1)-based variant).

Programs

Maple

Andre := proc(n, k) option remember; local j;
  ifelse(k = 0, 1, ifelse(n = 0, 1,
  -add(binomial(k, j) * Andre(n, j), j = 0..k-1, n))) end:
T := (n, k) -> abs(Andre(k, n)): seq(seq(T(n, k), k = 0..n), n = 0..10);

Mathematica

Andre[n_, k_] := Andre[n, k] = If[k <= 0, 1, If[n == 0, 1, -Sum[Binomial[k, j] Andre[n, j], {j, 0, k-1, n}]]];
(* Seen as an array: *)
A[n_, k_] := Abs[Andre[k, n + k]];
Table[A[n, k], {n, 0, 9}, {k, 0, 7}] // MatrixForm