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A126963
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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).
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2
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1, 5, 43, 177, 2867, 11531, 92479, 370345, 11857475, 47442055, 379582629, 1518418695, 24295375159, 97182800711, 777467420263, 3109879375897, 199032580597603, 796130905791967, 6369049515119561, 25476202478636219, 407619274119811709, 1630477163761481141
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OFFSET
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0,2
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LINKS
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FORMULA
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f(n) = Sum_{k=0..n} binomial(-1/2,k)*(-1/2)^k.
f(n) -> sqrt(2) as n -> oo.
G.f.: (sqrt(-x)*arccsc(1-x)/sqrt(2)-(Pi*i*sqrt(x))/sqrt(2)^3)/x. - Vladimir Kruchinin, Oct 10 2012
a(n) = numerator( Sum_{k=0..n} binomial(2*k, k)/8^k ). - G. C. Greubel, Jan 29 2020
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MAPLE
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seq( numer( add(binomial(2*k, k)/8^k, k=0..n) ), n=0..25); # G. C. Greubel, Jan 29 2020
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MATHEMATICA
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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 *)
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)]];
Table[Numerator[f[n]], {n, 0, 25}] (* G. C. Greubel, Jan 29 2020 *)
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PROG
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(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
(Magma) [Numerator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // G. C. Greubel, Jan 29 2020
(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
(GAP) List([0..25], n-> NumeratorRat( Sum([0..n], k-> Binomial(2*k, k)/8^k) )); # G. C. Greubel, Jan 29 2020
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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