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

1,5

Links

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

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)

Example

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;

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

Prog

(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

Crossrefs

Sequence in context: A193636 A232968 A119673 * A051455 A346875 A289511

Adjacent sequences: A144444 A144445 A144446 * A144448 A144449 A144450

Keyword

nonn,tabl

Author

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

Extensions

Edited by G. C. Greubel, Mar 06 2022

Status

approved