A144440 Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 11, 1, 1, 54, 54, 1, 1, 229, 789, 229, 1, 1, 932, 7975, 7975, 932, 1, 1, 3747, 68628, 161867, 68628, 3747, 1, 1, 15010, 543144, 2534759, 2534759, 543144, 15010, 1, 1, 60065, 4098439, 34243778, 66389335, 34243778, 4098439, 60065, 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,    11,       1;
  1,    54,      54,        1;
  1,   229,     789,      229,        1;
  1,   932,    7975,     7975,      932,        1;
  1,  3747,   68628,   161867,    68628,     3747,       1;
  1, 15010,  543144,  2534759,  2534759,   543144,   15010,     1;
  1, 60065, 4098439, 34243778, 66389335, 34243778, 4098439, 60065, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144441, A144442, A144443, A144444, A144445.

Programs

Mathematica

T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j] ];
Table[T[n,k,3,3], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)

Sage

def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144440(n,k): return T(n,k,3,3)
flatten([[A144440(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = s(n), where s(n) = (3*n-4)*s(n-1) + 3*s(n-2), s(1) = 1, s(2) = 2.

From G. C. Greubel, Mar 03 2022: (Start)

T(n, n-k) = T(n, k).

T(n, 2) = (1/3)*(11*4^(n-2) - (3*n+2)).

T(n, 3) = (1/18)*(9*n^2 + 3*n - 11 - 22*4^(n-3)*(12*n-1) + 709*7^(n-3)). (End)