cp's OEIS Frontend

A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

%I A349648 #30 Apr 28 2023 20:25:32
%S A349648 1,2,2,8,14,64,132,640,1430,7168,16796,86016,208012,1081344,2674440,
%T A349648 14057472,35357670,187432960,477638700,2549088256,6564120420,
%U A349648 35223764992,91482563640,493132709888,1289904147324,6979724509184,18367353072152,99710350131200
%N A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).
%F A349648 a(2*n) = A048990(n) = A000108(2*n), n>=0.
%F A349648 a(2*n+1) = A052707(n+1) = 2^(2*n+1)*A000108(n), n>=0.
%F A349648 G.f.: A(x) = C(x)/C(-x) = (1 - sqrt(1 - 4*x))/(sqrt(1 + 4*x) - 1), where C(x) is the g.f. of A000108.
%F A349648 G.f.: A(x) = F(x^2) + 2*x*F(x^2)^2 = (C(x) + C(-x))/2 + 2*x*C(4*x^2), where F(x) is the g.f. of A048990.
%F A349648 G.f.: A(-x) = 1/A(x).
%F A349648 G.f.: A(x) = R(x*C(-x)^2) = 1/R(-x*C(x)^2), where R(x) is the g.f. of A006318.
%F A349648 G.f.: A(x) = (1 + x*C(x)*C(-x))/(1 - x*C(x)*C(-x)), see A079489 for the expansion of C(x)*C(-x).
%F A349648 D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*(n-1)*(8*n^2-32*n+35)*a(n-2) +64*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Mar 06 2022
%F A349648 Sum_{n>=0} 1/a(n) = 28/15 + 2*Pi/(9*sqrt(3)) + 64*arcsin(1/4)/(75*sqrt(15)) - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). - _Amiram Eldar_, Apr 20 2023
%F A349648 G.f.: A(x) = exp( Sum_{n >= 1} binomial(4*n-2,2*n-1)*x^(2*n-1)/(2*n-1) ). - _Peter Bala_, Apr 28 2023
%p A349648 gf:= (c-> c(x)/c(-x))(x-> hypergeom([1/2, 1], [2], 4*x)):
%p A349648 a:= n-> coeff(series(gf, x, n+1), x, n):
%p A349648 seq(a(n), n=0..35);  # _Alois P. Heinz_, Nov 23 2021
%t A349648 CoefficientList[Series[(1-Sqrt[1-4x])/(Sqrt[1+4x]-1),{x,0,24}],x]
%Y A349648 Cf. A000108, A001622, A048990 (bijection), A052707 (bijection), A006318, A079489, A246062, A333564.
%K A349648 nonn,easy
%O A349648 0,2
%A A349648 _Alexander Burstein_, Nov 23 2021