A144444 Triangle read by rows: T(n, k) = (1-n+k)*T(n-1, k-1) + (2-k)*T(n-1, k) - T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, -1, 1, 1, -2, -2, 1, 1, -3, 5, -3, 1, 1, -4, 3, 3, -4, 1, 1, -5, 12, -17, 12, -5, 1, 1, -6, 12, -5, -5, 12, -6, 1, 1, -7, 23, -50, 47, -50, 23, -7, 1, 1, -8, 25, -27, 64, 64, -27, 25, -8, 1

Offset: 1

Author

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

Keywords

• sign
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1, -1,  1;
  1, -2, -2,   1;
  1, -3,  5,  -3,  1;
  1, -4,  3,   3, -4,   1;
  1, -5, 12, -17, 12,  -5,   1;
  1, -6, 12,  -5, -5,  12,  -6,  1;
  1, -7, 23, -50, 47, -50,  23, -7,  1;
  1, -8, 25, -27, 64,  64, -27, 25, -8, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144445.

Cf. A075374.

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,-1], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 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 A144444(n,k): return T(n,k,-1,-1)
flatten([[A144444(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Formula

T(n, k) = (1-n+k)*T(n-1, k-1) + (2-k)*T(n-1, k) - 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) = -(n-4)*s(n-1) - s(n-2), s(1) = 1, s(2) = 2.

From G. C. Greubel, Mar 04 2022: (Start)

Sum_{k=1..n} T(n, k) = 2*[n<3] + (-1)^(n-1)*A075374(n-2).

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

T(n, 2) = [n=2] - n + 2.

T(n, 3) = (1/2)*((n^2 -5*n +5) -5*(-1)^n) - [n=3]. (End)