OFFSET
0,2
FORMULA
G.f.: A(x) = -sqrt(1/2*(1+sqrt(1-16*x))).
With interpolated zeros, this has g.f. -(sqrt(1-4x)+sqrt(1+4x))/2. - Paul Barry, Dec 23 2006
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
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
a(n) = A001448(n)/(4*n-1). - R. J. Mathar, Apr 27 2020
From Peter Bala, Apr 02 2023: (Start)
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.
The series reversion of -x*A(x) is equal to x * the o.g.f. of A245112. (End)
EXAMPLE
sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 21/1024*x^3 - ...
MATHEMATICA
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 *)
PROG
(Magma) [(1/(4*n-1))*Binomial(4*n, 2*n) : n in [0..20]]; // Wesley Ivan Hurt, Jan 06 2024
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Mar 10 2013
STATUS
approved