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

Author

Gary W. Adamson, Aug 28 2007

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A003506, A066524.

Programs

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

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 *)

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

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)