A144447 Triangle T(n, k) = T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1), with T(n, 1) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 4, 1, 1, 7, 13, 1, 1, 10, 34, 49, 1, 1, 13, 64, 160, 211, 1, 1, 16, 103, 361, 781, 994, 1, 1, 19, 151, 679, 1981, 3967, 4963, 1, 1, 22, 208, 1141, 4162, 10891, 20815, 25780, 1, 1, 25, 274, 1774, 7756, 24790, 60463, 112021, 137803, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1,  7,  13,    1;
  1, 10,  34,   49,    1;
  1, 13,  64,  160,  211,     1;
  1, 16, 103,  361,  781,   994,     1;
  1, 19, 151,  679, 1981,  3967,  4963,      1;
  1, 22, 208, 1141, 4162, 10891, 20815,  25780,      1;
  1, 25, 274, 1774, 7756, 24790, 60463, 112021, 137803, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Programs

Mathematica

T[n_, k_]:= T[n,k]= If[k==1 || k==n, 1, T[n-1,k]+T[n,k-1]+T[n-1,k-1]+T[n-2,k-1]];
Table[T[n, k], {n,15}, {k,n}]//Flatten

Sage

def T(n,k): return 1 if (k==1 or k==n) else T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1) # A144447
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 06 2022

Formula

T(n, k) = T(n-1, k) + T(n, k-1) + T(n-1, k-1) + T(n-2, k-1), with T(n, 1) = T(n, n) = 1.

From G. C. Greubel, Mar 09 2022: (Start)

T(n, 2) = (3*n) - 5.

T(n, 3) = (1/2!)*((3*n)^2 - 13*(3*n) + 38).

T(n, 4) = (1/3!)*((3*n)^3 - 24*(3*n)^2 + 195*(3*n) - 606).

T(n, 5) = (1/4!)*((3*n)^4 - 38*(3*n)^3 + 579*(3*n)^2 - 4422*(3*n) + 13704). (End)

Extensions

Edited by G. C. Greubel, Mar 06 2022