A174846 E.g.f.: AGM(1, exp(4x)), where AGM(x, y) is the arithmetic-geometric mean of Gauss.

1, 2, 6, 20, 66, 212, 756, 3320, 11346, -11068, 14556, 7202120, 18928476, -1376971048, -3526491144, 394396083920, 1016723438706, -148493230507228, -383613651929844, 71479338751223720, 184867683069498036

Offset

0,2

Comments

Conjecture: limit |a(n)/n!|^(-1/n) = r exists and is finite with r<0.8...

What is the radius of convergence of the e.g.f. as a power series in x?

r = Pi/4. - Vaclav Kotesovec, Sep 27 2019

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Formula

E.g.f.: exp(2x)*AGM(1, cosh(2x)).

E.g.f.: exp(2x)*AGM( cosh(x)^2, sqrt(cosh(2x)) ).

Example

E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 20*x^3/3! + 66*x^4/4! +...
Special value:
A(log(2)/8) = Pi^(3/2)*sqrt(8)/gamma(1/4)^2 = 1.19814023473...

Mathematica

nmax = 20; CoefficientList[Series[E^(4*x)*Pi / (2*EllipticK[1 - E^(-8*x)]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 27 2019 *)

Prog

(PARI) {a(n)=n!*polcoeff(agm(1, exp(4*x+x*O(x^n))), n)}

Crossrefs

Sequence in context: A279460 A096487 A083323 * A369431 A111285 A052991

Adjacent sequences: A174843 A174844 A174845 * A174847 A174848 A174849

Keyword

sign

Author

Paul D. Hanna, Jan 24 2011

Status

approved