A132751 Triangle T(n, k) = 2/Beta(n-k+1, k) - 1, read by rows.

1, 3, 3, 5, 11, 5, 7, 23, 23, 7, 9, 39, 59, 39, 9, 11, 59, 119, 119, 59, 11, 13, 83, 209, 279, 209, 83, 13, 15, 111, 335, 559, 559, 335, 111, 15, 17, 143, 503, 1007, 1259, 1007, 503, 143, 17, 19, 179, 719, 1679, 2519, 2519, 1679, 719, 179, 19

Offset

1,2

Links

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Formula

T(n, k) = 2*A003506(n, k) - 1, an infinite lower triangular matrix.

From G. C. Greubel, Feb 16 2021: (Start)

T(n, k) = 2/Beta(n-k+1, k) - 1.

Sum_{k=1..n} T(n, k) = n*(2^n -1) = A066524(n). (End)

Example

First few rows of the triangle are:
   1;
   3,   3;
   5,  11,   5;
   7,  23,  23,   7;
   9,  39,  59,  39,   9;
  11,  59, 119, 119,  59,  11;
  13,  83, 209, 279, 209,  83,  13;
  15, 111, 335, 559, 559, 335, 111, 15;
  ...

Mathematica

T[n_, k_]:= 2/Beta[n-k+1, k] - 1;
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

Prog

(Sage)
def A132751(n, k): return 2/beta(n-k+1, k) - 1
flatten([[A132751(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 16 2021
(Magma)
A132751:= func< n, k | 2*Factorial(n)/(Factorial(k-1)*Factorial(n-k)) -1 >;
[A132751(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 16 2021

Crossrefs

Cf. A003506, A066524.

Sequence in context: A326175 A072337 A335056 * A218354 A286514 A235859

Adjacent sequences: A132748 A132749 A132750 * A132752 A132753 A132754

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 28 2007

Status

approved