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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Row sums are A151798.

Sequence in context: A316863 A037911 A176046 * A037912 A056980 A268643

Adjacent sequences: A172087 A172088 A172089 * A172091 A172092 A172093

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Apr 29 2021

Status

approved