A246062 -id:A246062 - cp's OEIS frontend

A193618 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^-2 - A(-x)^-2 = -8*x.

1, 2, -2, -28, 54, 860, -2004, -33720, 86054, 1492908, -4019452, -71101832, 198310460, 3555617432, -10168382696, -184127171952, 536496907782, 9788598556876, -28937139277804, -531135371147368, 1588378827366868, 29295861148032584

Offset: 0

Paul D. Hanna, Aug 01 2011

sign

The unsigned version of this sequence, A246062, has g.f.: sqrt( (1 + sqrt(1+8*x)) / (1 + sqrt(1-8*x)) ).

			G.f.: A(x) = 1 + 2*x - 2*x^2 - 28*x^3 + 54*x^4 + 860*x^5 - 2004*x^6 +...
where
A(x)^2 = 1 + 4*x - 64*x^3 + 2048*x^5 - 81920*x^7 + 3670016*x^9 +...
and
A(x)^-2 = 1 - 4*x + 16*x^2 - 256*x^4 + 8192*x^6 - 327680*x^8 +...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A193619, A246062.

{a(n)=local(Ox=x*O(x^n),A=(2*(sqrt(1+64*x^2+Ox)+8*x)/(sqrt(1+64*x^2+Ox)+1))^(1/4));polcoeff(A,n)}

N=40; x='x+O('x^N); Vec(sqrt(2/(1-8*x+sqrt(1+64*x^2)))) \\ Seiichi Manyama, Aug 26 2020

G.f.: ( 2*(sqrt(1+64*x^2) + 8*x)/(sqrt(1+64*x^2) + 1) )^(1/4).

G.f. A(x) = 1/G(x) where G(x) is the g.f. of A193619.

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

Original entry on oeis.org

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

Alexander Burstein, Nov 23 2021

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

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

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

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

cp's OEIS Frontend

A193618 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^-2 - A(-x)^-2 = -8*x.

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