A100512 Numerator of Sum_{k=0..n} 1/C(2*n, 2*k).

1, 2, 13, 32, 73, 647, 28211, 6080, 18181, 1542158, 2786599, 29229544, 134354573, 745984697, 80530073893, 291816652544, 274050911261, 258328905974, 18079412000719, 8574689239808, 334365081328507, 13707288497202919, 52386756782140399, 589296748617180608

Offset: 0

Author

N. J. A. Sloane, Nov 25 2004

Keywords

• nonn
• frac

Examples

			Sum_{k=0..n} 1/binomial(2*n, 2*k) = {1, 2, 13/6, 32/15, 73/35, 647/315, 28211/13860, 6080/3003, 18181/9009, 1542158/765765, 2786599/1385670, 29229544/14549535, 134354573/66927861, ...} = a(n)/A100513(n).
		

References

• M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A046825, A100513, A100514, A100515.

Programs

Magma

[Numerator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023

Mathematica

Table[Sum[1/Binomial[2n,2k],{k,0,n}],{n,0,30}]//Numerator (* Harvey P. Dale, Aug 12 2016 *)

SageMath

def A100512(n): return numerator((2*n+1)*sum(beta(2*k+1, 2*n-2*k+1) for k in range(n+1)))
[A100512(n) for n in range(40)] # G. C. Greubel, Mar 28 2023

Formula

a(n) = numerator( Sum_{k=0..n} 1/binomial(2*n, 2*k) ).

a(n) = numerator( (2*n+1)*Sum_{k=0..n} beta(2*k+1, 2*n-2*k+1) ). - G. C. Greubel, Mar 28 2023