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

1, 2, 2, 8, 14, 64, 132, 640, 1430, 7168, 16796, 86016, 208012, 1081344, 2674440, 14057472, 35357670, 187432960, 477638700, 2549088256, 6564120420, 35223764992, 91482563640, 493132709888, 1289904147324, 6979724509184, 18367353072152, 99710350131200

Offset: 0

Author

Alexander Burstein, Nov 23 2021

Keywords

• nonn
• easy

Crossrefs

Cf. A000108, A001622, A048990 (bijection), A052707 (bijection), A006318, A079489, A246062, A333564.

Programs

Maple

gf:= (c-> c(x)/c(-x))(x-> hypergeom([1/2, 1], [2], 4*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 23 2021

Mathematica

CoefficientList[Series[(1-Sqrt[1-4x])/(Sqrt[1+4x]-1),{x,0,24}],x]

Formula

a(2*n) = A048990(n) = A000108(2*n), n>=0.

a(2*n+1) = A052707(n+1) = 2^(2*n+1)*A000108(n), n>=0.

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.

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.

G.f.: A(-x) = 1/A(x).

G.f.: A(x) = R(x*C(-x)^2) = 1/R(-x*C(x)^2), where R(x) is the g.f. of A006318.

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).

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

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

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