A024491 - OEIS

%I #22 Jan 06 2024 21:37:09

%S -1,2,10,84,858,9724,117572,1485800,19389690,259289580,3534526380,

%T 48932534040,686119227300,9723892802904,139067101832008,

%U 2004484433302736,29089272078453818,424672260824486220,6232570989814602524,91901608649243484728,1360850743459951600780

%N a(n) = (1/(4n-1))*C(4n,2n).

%F G.f.: A(x) = -sqrt(1/2*(1+sqrt(1-16*x))).

%F With interpolated zeros, this has g.f. -(sqrt(1-4x)+sqrt(1+4x))/2. - _Paul Barry_, Dec 23 2006

%F D-finite with recurrence n*(2*n-1)*a(n) -2*(4*n-3)*(4*n-5)*a(n-1)=0. - _R. J. Mathar_, Nov 13 2012

%F G.f.: -1/2*G(0), where G(k)= 1 + 1/(1 - 2*sqrt(x)*(4*k-1)/(2*sqrt(x)*(4*k-1) + (2*k+1)/(1 - 1/(1 - sqrt(x)*(4*k+1)/(sqrt(x)*(4*k+1) - (k+1)/G(k+1) ))))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 19 2013

%F a(n) = A001448(n)/(4*n-1). - _R. J. Mathar_, Apr 27 2020

%F From _Peter Bala_, Apr 02 2023: (Start)

%F O.g.f. A(x) = - sqrt(1 - 4*x*C(4*x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

%F The series reversion of -x*A(x) is equal to x * the o.g.f. of A245112. (End)

%e sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 21/1024*x^3 - ...

%t Table[1/(4n-1) Binomial[4n,2n],{n,0,20}] (* or *) With[{c=4Sqrt[x]}, CoefficientList[ Series[(-Sqrt[1-c]-Sqrt[1+c])/2,{x,0,30}],x]] (* _Harvey P. Dale_, Mar 10 2013 *)

%o (Magma) [(1/(4*n-1))*Binomial(4*n,2*n) : n in [0..20]]; // _Wesley Ivan Hurt_, Jan 06 2024

%Y Cf. A000108, A024492, A245112.

%K sign,easy

%O 0,2

%A _Clark Kimberling_

%E More terms from _Harvey P. Dale_, Mar 10 2013