This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A024491 #33 Sep 04 2025 02:34:00
%S A024491 -1,2,10,84,858,9724,117572,1485800,19389690,259289580,3534526380,
%T A024491 48932534040,686119227300,9723892802904,139067101832008,
%U A024491 2004484433302736,29089272078453818,424672260824486220,6232570989814602524,91901608649243484728,1360850743459951600780
%N A024491 a(n) = (1/(4n-1))*C(4n,2n).
%F A024491 G.f.: A(x) = -sqrt((1/2)*(1+sqrt(1-16*x))).
%F A024491 With interpolated zeros, this has g.f. -(sqrt(1-4x)+sqrt(1+4x))/2. - _Paul Barry_, Dec 23 2006
%F A024491 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 A024491 a(n) = A001448(n)/(4*n-1). - _R. J. Mathar_, Apr 27 2020
%F A024491 From _Peter Bala_, Apr 02 2023: (Start)
%F A024491 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 A024491 The series reversion of -x*A(x) is equal to x * the o.g.f. of A245112. (End)
%F A024491 a(n) ~ 2^(4*n-5/2) / (n^(3/2) * sqrt(Pi)). - _Amiram Eldar_, Sep 04 2025
%e A024491 sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 21/1024*x^3 - ...
%t A024491 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 A024491 (Magma) [(1/(4*n-1))*Binomial(4*n,2*n) : n in [0..20]]; // _Wesley Ivan Hurt_, Jan 06 2024
%Y A024491 Cf. A000108, A001448, A024492, A245112.
%K A024491 sign,easy,changed
%O A024491 0,2
%A A024491 _Clark Kimberling_
%E A024491 More terms from _Harvey P. Dale_, Mar 10 2013