A100513 Denominator of Sum_{k=0..n} 1/C(2*n,2*k).

1, 1, 6, 15, 35, 315, 13860, 3003, 9009, 765765, 1385670, 14549535, 66927861, 371821450, 40156716600, 145568097675, 136745788725, 128931743655, 9025222055850, 4281195077775, 166966608033225, 6845630929362225, 26165522663340060, 294362129962575675

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, ...} = A100512(n)/a(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, A100512, A100514, A100515.

Programs

Magma

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

Mathematica

Table[Denominator[(2*n+1)*Sum[Beta[2k+1,2(n-k)+1], {k,0,n}]], {n,0,40}] (* G. C. Greubel, Mar 28 2023 *)

SageMath

def A100513(n): return denominator((2*n+1)*sum(beta(2*k+1, 2*(n-k)+1) for k in range(n+1)))
[A100513(n) for n in range(40)] # G. C. Greubel, Mar 28 2023

Formula

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

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