OFFSET
0,3
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
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
EXAMPLE
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).
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 *)
PROG
(Magma) [Denominator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Nov 25 2004
STATUS
approved