OFFSET
0,4
COMMENTS
Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
Limit a(n+1)/a(n) = 3.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 +...
As a logarithmic expansion,
L(x) = x + x^2/2 + x^3/3 + 9*x^4/4 + 41*x^5/5 + 121*x^6/6 + 281*x^7/7 + 673*x^8/8 + 2017*x^9/9 + 6721*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...
equals 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
a(n) ~ 3^(n+1) / (2*log(n)) * (1 + (log(3) - 3*log(2) - gamma) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 30 2019
MATHEMATICA
Simplify[CoefficientList[Series[D[Log[Sqrt[(2*EllipticK[1 - (1 - 3*x)^4/(1 + x)^4])/Pi] / (1 + x)], x], {x, 0, 30}], x]] (* Vaclav Kotesovec, Sep 27 2019 *)
PROG
(PARI) /* As the logarithmic derivative of A245931: */
{a(n)=local(G=1); G = 1 / sqrt( agm((1-3*x)^2, (1+x)^2 +x^2*O(x^n)) ); polcoeff(G'/G, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) /* As the logarithm of g.f. of A245931 (offset = 1): */
{a(n)=local(A=1); A = -log( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) )/2; n*polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 14 2014
STATUS
approved