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%I #24 Sep 08 2022 08:45:29
%S 1,5,43,177,2867,11531,92479,370345,11857475,47442055,379582629,
%T 1518418695,24295375159,97182800711,777467420263,3109879375897,
%U 199032580597603,796130905791967,6369049515119561,25476202478636219,407619274119811709,1630477163761481141
%N Numerators of sequence defined by f(0)=1, f(1)=5/4; f(n) = ( (6*n-1)*f(n-1) - (2*n-1)*f(n-2) )/(4n).
%H G. C. Greubel, <a href="/A126963/b126963.txt">Table of n, a(n) for n = 0..500</a>
%H D. Doster, <a href="http://www.jstor.org/stable/2691073">Problem 1318, Three Term Recurrence</a>, Math. Magazine, 63 (1990), 127-128.
%F f(n) = Sum_{k=0..n} binomial(-1/2,k)*(-1/2)^k.
%F f(n) -> sqrt(2) as n -> oo.
%F G.f.: (sqrt(-x)*arccsc(1-x)/sqrt(2)-(Pi*i*sqrt(x))/sqrt(2)^3)/x. - _Vladimir Kruchinin_, Oct 10 2012
%F a(n) = numerator( Sum_{k=0..n} binomial(2*k, k)/8^k ). - _G. C. Greubel_, Jan 29 2020
%p seq( numer( add(binomial(2*k, k)/8^k, k=0..n) ), n=0..25); # _G. C. Greubel_, Jan 29 2020
%t a[n_] := Sqrt[2](1-(Gamma[1/2+n] Hypergeometric2F1[n,1/2+n,1+n,-1])/(Sqrt[Pi] Gamma[1+n])); Table[Numerator[FullSimplify[a[n]]], {n,20}] (* _Gerry Martens_, Aug 09 2015 *)
%t f[n_]:= If[n==0, 1, If[n==1, 5/4, ((6*n-1)*f[n-1]-(2*n-1)*f[n-2])/(4*n)]];
%t Table[Numerator[f[n]], {n, 0, 25}] (* _G. C. Greubel_, Jan 29 2020 *)
%o (PARI) A126963(n)=numerator(sum(k=0,n,binomial(-1/2,k)/(-2)^k)) \\ f(n)=if(n>1,((6*n-1)*f(n-1)-(2*n-1)*f(n-2))/(4*n),(5/4)^n) yields the same results. - _M. F. Hasler_, Aug 11 2015
%o (Magma) [Numerator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // _G. C. Greubel_, Jan 29 2020
%o (Sage) [numerator( sum(binomial(2*k, k)/8^k for k in (0..n)) ) for n in (0..25)] # _G. C. Greubel_, Jan 29 2020
%o (GAP) List([0..25], n-> NumeratorRat( Sum([0..n], k-> Binomial(2*k,k)/8^k) )); # _G. C. Greubel_, Jan 29 2020
%Y Denominators are in A088802.
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_, Mar 20 2007
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