A249520 - OEIS

%I #37 Nov 11 2024 22:23:49

%S 1,5,17,134,1333,14890,178394,2240268,29101197,387793090,5271628846,

%T 72818272852,1019157450818,14421479205284,205978607191508,

%U 2965567162041368,42994130150806077,627124332791860146,9196734644381065510,135516564162069215748,2005456310676742385910

%N Expansion of: G.f.: 2-(2*x)/(1-sqrt(sqrt(1-16*x)+1)/sqrt(2)).

%H G. C. Greubel, <a href="/A249520/b249520.txt">Table of n, a(n) for n = 0..830</a>

%F a(n) = (Sum_{i=0..n} 2^i*binomial(2*(n-1)+i-1,i)*binomial(2*n-i-2,n-i))/(n-1), n>1, a(0)=1, a(1)=5.

%F a(n) ~ (3+2*sqrt(2)) * 2^(4*n-9/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Oct 31 2014

%F D-finite with recurrence: n*(n-1)*(2*n-3)*a(n) -2*(n-1)*(32*n^2-128*n+135)*a(n-1) +16*(2*n-5)*(4*n-9)*(4*n-11)*a(n-2)=0. - _R. J. Mathar_, Jan 25 2020

%F a(n) = 2^(2*n-1)*C(n-1) + C(2*n-1) + 2*C(2*n-2), for n>0, where C(n) is the n-th Catalan number, A000108. - _Ira M. Gessel_, Dec 10 2020

%F a(n) = binomial(2*n - 2, n)*hypergeom([-n, 2*n - 2], [2 - 2*n], 2)/(n - 1) for n >= 2. - _Peter Luschny_, Dec 10 2020

%p a := n -> `if`(n=0,1,`if`(n=1,5,((4^(2*n-1))/((+2*n-1)*(4*n-3)*(4*n-1) *Pi*GAMMA(1+2*n)))*((6*sqrt(Pi)*(1-2*n)^2*GAMMA(2*n+1/2)+4^n*(4*n-3)*(4*n-1)*GAMMA(n+1/2)^2)))): seq(a(n),n=0..18); # _Peter Luschny_, Oct 31 2014

%t CoefficientList[Series[2-(2*x)/(1-Sqrt[Sqrt[1-16*x]+1]/Sqrt[2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 31 2014 *)

%t a[n_] := (Binomial[2n - 2, n] Hypergeometric2F1[-n, 2n - 2, 2 - 2n, 2])/(n - 1);

%t a[0] := 1; a[1] := 5; Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Dec 10 2020 *)

%o (Maxima)

%o a(n)=if n=0 then 1 else if n=1 then 5 else sum(2^i*binomial(2*(n-1)+i-1,i)*binomial(2*n-i-2,n-i),i,0,n)/(n-1);

%o (PARI) my(x='x+O('x^50)); Vec(2 - (2*x)/(1-sqrt(sqrt(1-16*x)+1)/sqrt(2))) \\ _G. C. Greubel_, Jun 02 2017

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Oct 31 2014