A144436 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.

1, 1, 1, 1, 8, 1, 1, 23, 23, 1, 1, 54, 170, 54, 1, 1, 117, 818, 818, 117, 1, 1, 244, 3255, 7224, 3255, 244, 1, 1, 499, 11697, 48443, 48443, 11697, 499, 1, 1, 1010, 39560, 276974, 513326, 276974, 39560, 1010, 1, 1, 2033, 128756, 1431604, 4422246

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 04 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    8,      1;
  1,   23,     23,       1;
  1,   54,    170,      54,       1;
  1,  117,    818,     818,     117,       1;
  1,  244,   3255,    7224,    3255,     244,       1;
  1,  499,  11697,   48443,   48443,   11697,     499,      1;
  1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;
  1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144435.

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,1,4], {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 A144436(n,k): return T(n,k,1,4)
flatten([[A144436(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4.

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

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

T(n, 2) = 2^(n+1) - (n+5).

T(n, 3) = (1/2)*( n^2 + 9*n + 16 - 2^(n+2)*(n+3) + 142*3^(n-3) ). (End)

Extensions

Edited by G. C. Greubel, Mar 03 2022