A126963 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).

1, 5, 43, 177, 2867, 11531, 92479, 370345, 11857475, 47442055, 379582629, 1518418695, 24295375159, 97182800711, 777467420263, 3109879375897, 199032580597603, 796130905791967, 6369049515119561, 25476202478636219, 407619274119811709, 1630477163761481141

Offset: 0

Author

N. J. A. Sloane, Mar 20 2007

Keywords

• nonn
• frac

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• D. Doster, Problem 1318, Three Term Recurrence, Math. Magazine, 63 (1990), 127-128.

Crossrefs

Denominators are in A088802.

Programs

GAP

List([0..25], n-> NumeratorRat( Sum([0..n], k-> Binomial(2*k,k)/8^k) ));  # G. C. Greubel, Jan 29 2020

Magma

[Numerator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // G. C. Greubel, Jan 29 2020

Maple

seq( numer( add(binomial(2*k, k)/8^k, k=0..n) ), n=0..25); # G. C. Greubel, Jan 29 2020

Mathematica

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 *)

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

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

Formula

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