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

0,5

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).

Links

Table of n, a(n) for n=0..65.

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

Example

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;
  ...

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 *)

Crossrefs

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

Sequence in context: A141523 A285808 A201588 * A086385 A295222 A362078

Adjacent sequences: A336855 A336856 A336857 * A336859 A336860 A336861

Keyword

nonn,tabl

Author

Petros Hadjicostas, Aug 05 2020

Status

approved