A172090 Triangle T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2, read by rows.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Author

Roger L. Bagula, Jan 25 2010

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 1, 1;
  1, 1, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Row sums are A151798.

Programs

Mathematica

(* First program *)
f[n_]:= f[n]= If[n < 2, (-1)^n*(n+1), -3*n];
T[n_, k_]:= f[n-k] +f[k] -f[n];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 29 2021 *)
(* Second program *)
T[n_, k_]:= If[n<3, Binomial[n, k], If[n==3 || k<2 || k>n-2, 1, 0]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2021 *)

Sage

def f(n): return (-1)^n*(n+1) if (n<2) else -3*n
def T(n,k): return f(n-k) + f(k) - f(n)
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 29 2021

Formula

T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2.

From G. C. Greubel, Apr 29 2021: (Start)

T(n, k) is defined by T(n, 0) = T(n, 1) = T(n, n-1) = T(n, n) = T(3, k) = 1, T(2, 1) = 2 and 0 otherwise.

Sum_{k=0..n} T(n,k) = A151798(n). (End)

Extensions

Edited by G. C. Greubel, Apr 29 2021