A144432 Triangle, T(n, k), read by rows: T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m*(n-k) + 1)*t(n-1, k-1) + (m*k - (m-1))*t(n-1, k) and m = -1.

-1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 5, 1, 5, -1, -1, 11, 1, 1, 11, -1, -1, 19, 41, 71, 41, 19, -1, -1, 29, 71, 29, 29, 71, 29, -1, -1, 41, 239, 701, 869, 701, 239, 41, -1, -1, 55, 379, 811, 181, 181, 811, 379, 55, -1

Offset: 1

Author

Roger L. Bagula, Oct 04 2008

Keywords

• sign
• tabl

Examples

			Triangle begins as:
  -1;
  -1, -1;
  -1, -1,  -1;
  -1,  1,   1,  -1;
  -1,  5,   1,   5,  -1;
  -1, 11,   1,   1,  11,  -1;
  -1, 19,  41,  71,  41,  19,  -1;
  -1, 29,  71,  29,  29,  71,  29,  -1;
  -1, 41, 239, 701, 869, 701, 239,  41, -1;
  -1, 55, 379, 811, 181, 181, 811, 379, 55, -1;
		

Links

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

Crossrefs

Cf. A098593, A144431.

Programs

Mathematica

t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*(n-k)+1)*t[n-1,k-1,m] + (m*(k - 1)+1)*t[n-1,k,m]];
T[n_, k_, m_]:= t[n,k,m]^2 -t[n,k,m] -1;
Table[T[n,k,-1], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2022 *)

Sage

def t(n,k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
def A144432(n,k): return t(n,k)^2 - t(n,k) - 1
flatten([[A144432(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 02 2022

Formula

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

From G. C. Greubel, Mar 02 2022: (Start)

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

T(n, k) = t(n,k)^2 - t(n,k) - 1, where t(n,k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with t(1, k) = t(2, k) = 1.

Sum_{k=1..n} T(n,k) = -n*[n<4] + ( 2*binomial(2*n-6, n-3)*(binomial(n-1,2) - (-1)^n*binomial(n-3,2))/binomial(n-1,2) - n )*[n>=4]. (End)

Extensions

Edited by G. C. Greubel, Mar 02 2022