A141901 Triangle T(n, k) = Sum_{j=0..n-k-1} binomial(n, j+k+1) - 2^(n-k) with T(n, 0) = 1, read by rows.

1, 1, -1, 1, -1, -1, 1, 0, -1, -1, 1, 3, 1, -1, -1, 1, 10, 8, 2, -1, -1, 1, 25, 26, 14, 3, -1, -1, 1, 56, 67, 48, 21, 4, -1, -1, 1, 119, 155, 131, 77, 29, 5, -1, -1, 1, 246, 338, 318, 224, 114, 38, 6, -1, -1, 1, 501, 712, 720, 574, 354, 160, 48, 7, -1, -1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 13 2008

Keywords

• sign
• tabl

Examples

			Triangle begins as:
  1;
  1,  -1;
  1,  -1,  -1;
  1,   0,  -1,  -1;
  1,   3,   1,  -1,  -1;
  1,  10,   8,   2,  -1,  -1;
  1,  25,  26,  14,   3,  -1,  -1;
  1,  56,  67,  48,  21,   4,  -1, -1;
  1, 119, 155, 131,  77,  29,   5, -1, -1;
  1, 246, 338, 318, 224, 114,  38,  6, -1, -1;
  1, 501, 712, 720, 574, 354, 160, 48,  7, -1, -1;
		

Links

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

Crossrefs

Cf. A000225, A036799.

Programs

Magma

T:= func< n,k | k eq 0 select 1 else (&+[Binomial(n, j+k+1): j in [0..n]]) - 2^(n-k)>;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 29 2021

Mathematica

(* First program *)
T[n_, k_]:= If[k==0, 1, Binomial[n,k+1]*Hypergeometric2F1[1, 1+k-n, 2+k, -1] - 2^(n-k)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 29 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Sum[Binomial[n, j+k+1], {j, 0, n-k-1}] - 2^(n-k)];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 29 2021 *)

SageMath

@CachedFunction
def T(n,k):
    if k==0: return 1
    else: return sum( binomial(n, j+k+1) for j in (0..n-k-1) ) - 2^(n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2021

Formula

T(n, k) = binomial(n, k+1)*Hypergeometric2F1([1, 1+k-n], [2+k], -1) - 2^(n-k) with T(n, 0) = 1.

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

T(n, k) = Sum_{j=0..n-k-1} binomial(n, j+k+1) - 2^(n-k) with T(n, 0) = 1.

Sum_{k=0..n} T(n, k) = 2^(n-1)*(n-4) + 3 = A036799(n-3) - A000225(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 29 2021