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A100512 Numerator of Sum_{k=0..n} 1/C(2*n, 2*k). 3
1, 2, 13, 32, 73, 647, 28211, 6080, 18181, 1542158, 2786599, 29229544, 134354573, 745984697, 80530073893, 291816652544, 274050911261, 258328905974, 18079412000719, 8574689239808, 334365081328507, 13707288497202919, 52386756782140399, 589296748617180608 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.
LINKS
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
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, ...} = a(n)/A100513(n).
MATHEMATICA
Table[Sum[1/Binomial[2n, 2k], {k, 0, n}], {n, 0, 30}]//Numerator (* Harvey P. Dale, Aug 12 2016 *)
PROG
(Magma) [Numerator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023
(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
CROSSREFS
Sequence in context: A177455 A185950 A084828 * A051474 A062708 A296293
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Nov 25 2004
STATUS
approved

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Last modified April 15 00:51 EDT 2024. Contains 371667 sequences. (Running on oeis4.)