A144435 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 = 2, read by rows.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, -1, 0, 0, -1, 1, 1, -2, 3, 4, 3, -2, 1, 1, -3, 3, -17, -17, 3, -3, 1, 1, -4, 8, 28, 110, 28, 8, -4, 1, 1, -5, 10, -90, -476, -476, -90, 10, -5, 1

Offset: 1

Author

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

Keywords

• sign
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  1,  1,   1;
  1,  0,  2,   0,    1;
  1, -1,  0,   0,   -1,    1;
  1, -2,  3,   4,    3,   -2,   1;
  1, -3,  3, -17,  -17,    3,  -3,  1;
  1, -4,  8,  28,  110,   28,   8, -4,  1;
  1, -5, 10, -90, -476, -476, -90, 10, -5, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144436.

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,-1,2], {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 A144435(n,k): return T(n,k,-1,2)
flatten([[A144435(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 = 2.

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

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

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

T(n, 3) = (1/2)*(n^2 - 11*n + 32) - (-1)^n, n >= 4.

T(n, 4) = (1/12)*(-2*n^3 + 36*n^2 - 226*n + 496 - (-1)^n*(2^n - 12*(n-1))), n >= 5. (End)

Extensions

Edited by G. C. Greubel, Mar 03 2022