A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 5, 9, 1, 1, 7, 21, 31, 1, 1, 9, 37, 89, 121, 1, 1, 11, 57, 183, 393, 515, 1, 1, 13, 81, 321, 897, 1805, 2321, 1, 1, 15, 109, 511, 1729, 4431, 8557, 10879, 1, 1, 17, 141, 761, 3001, 9161, 22149, 41585, 52465, 1, 1, 19, 177, 1079, 4841, 17003, 48313, 112047, 206097, 258563, 1

Offset: 0

Author

Petros Hadjicostas, Aug 05 2020

Keywords

• nonn
• tabl

Comments

This is J. M. Bergot's triangular array described in A104858 with the top vertex of the triangle shifted from (1,1) to (0,0).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   9,   1;
  1,  7,  21,  31,    1;
  1,  9,  37,  89,  121,    1;
  1, 11,  57, 183,  393,  515,    1;
  1, 13,  81, 321,  897, 1805, 2321,     1;
  1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;
  ...
		

Crossrefs

Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.

Programs

Maple

A336858row := proc(n) option remember; local T, k, row;
row := Array(0..n, fill=1);
if n = 0 then return row fi; T := procname(n-1);
for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:
T := (n, k) -> A336858row(n)[k]:
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 06 2020

Mathematica

T[, 0] = 1; T[n, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

Formula

T(n,k) = T(n, k-1) + T(n-1, k) + T(n-1, k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.

T(n,k) = D(n,k) - Sum_{m=1..k} b(m-1)*D(n-m, k-m) - Sum_{m=0..k-1} D(n-m, k-m-1), where D(n,k) = A008288(n,k) (square array of Delannoy numbers) and b(n) = A086616(n).

T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.

T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.

T(n,n-1) = A086616(n-1) for n >= 1.

T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.

Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.

Bivariate o.g.f.: (1 - y - x*y*(1 + g(x*y)))/((1 - x*y)*(1 - x - y - x*y)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).

Bivariate o.g.f.: (1 - y - 2*x*y*q(x*y))/((1 - x*y)*(1 - x - y - x*y)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).

T(2*n,n) = A333090(n). - Peter Luschny, Aug 06 2020