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%I #7 Mar 31 2023 02:53:55
%S 1,1,6,15,35,315,13860,3003,9009,765765,1385670,14549535,66927861,
%T 371821450,40156716600,145568097675,136745788725,128931743655,
%U 9025222055850,4281195077775,166966608033225,6845630929362225,26165522663340060,294362129962575675
%N Denominator of Sum_{k=0..n} 1/C(2*n,2*k).
%D M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.
%H G. C. Greubel, <a href="/A100513/b100513.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = denominator( Sum_{k=0..n} 1/binomial(2*n,2*k) ).
%F a(n) = denominator( (2*n+1)*Sum_{k=0..n} beta(2*k+1, 2*(n-k)+1) ). - _G. C. Greubel_, Mar 28 2023
%e 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).
%t 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 *)
%o (Magma) [Denominator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // _G. C. Greubel_, Mar 28 2023
%o (SageMath)
%o def A100513(n): return denominator((2*n+1)*sum(beta(2*k+1, 2*(n-k)+1) for k in range(n+1)))
%o [A100513(n) for n in range(40)] # _G. C. Greubel_, Mar 28 2023
%Y Cf. A046825, A100512, A100514, A100515.
%K nonn,frac
%O 0,3
%A _N. J. A. Sloane_, Nov 25 2004
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