A132733 Triangle T(n, k) = 4*binomial(n, k) - 5 with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 11, 19, 11, 1, 1, 15, 35, 35, 15, 1, 1, 19, 55, 75, 55, 19, 1, 1, 23, 79, 135, 135, 79, 23, 1, 1, 27, 107, 219, 275, 219, 107, 27, 1, 1, 31, 139, 331, 499, 499, 331, 139, 31, 1, 1, 35, 175, 475, 835, 1003, 835, 475, 175, 35, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

T(n, k) = 2*A132731 - A000012, an infinite lower triangular matrix.

From G. C. Greubel, Feb 14 2021: (Start)

T(n, k) = 4*binomial(n, k) - 5 with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2^(n + 2) - (5*n + 1) - 2*[n=0] = A132734(n). (End)

Example

First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  7,  7,  1;
  1, 11, 19, 11,  1;
  1, 15, 35, 35, 15,  1;
  1, 19, 55, 75, 55, 19, 1;
  ...

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, 4*Binomial[n, k] - 5];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

Prog

(PARI) t(n, k) =  4*binomial(n, k) + 2*((k==0) || (k==n)) - 5*(k<=n); \\ Michel Marcus, Feb 12 2014
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else 4*binomial(n, k) - 5
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else 4*Binomial(n, k) - 5 >;
[T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

Crossrefs

Cf. A132731, A132734.

Sequence in context: A357940 A133800 A146900 * A347971 A082039 A367505

Adjacent sequences: A132730 A132731 A132732 * A132734 A132735 A132736

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 26 2007

Extensions

More terms from Michel Marcus, Feb 12 2014

Status

approved