|
%I #9 Mar 31 2023 05:12:39
%S 1,2,13,32,73,647,28211,6080,18181,1542158,2786599,29229544,134354573,
%T 745984697,80530073893,291816652544,274050911261,258328905974,
%U 18079412000719,8574689239808,334365081328507,13707288497202919,52386756782140399,589296748617180608
%N Numerator 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="/A100512/b100512.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = numerator( Sum_{k=0..n} 1/binomial(2*n, 2*k) ).
%F 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
%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, ...} = a(n)/A100513(n).
%t Table[Sum[1/Binomial[2n,2k],{k,0,n}],{n,0,30}]//Numerator (* _Harvey P. Dale_, Aug 12 2016 *)
%o (Magma) [Numerator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // _G. C. Greubel_, Mar 28 2023
%o (SageMath)
%o def A100512(n): return numerator((2*n+1)*sum(beta(2*k+1, 2*n-2*k+1) for k in range(n+1)))
%o [A100512(n) for n in range(40)] # _G. C. Greubel_, Mar 28 2023
%Y Cf. A046825, A100513, A100514, A100515.
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_, Nov 25 2004
|
