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A245931
G.f.: 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
3
1, 1, 1, 1, 3, 11, 31, 71, 157, 397, 1141, 3301, 9079, 24207, 65339, 182131, 517307, 1467067, 4128859, 11606683, 32835433, 93588097, 267745149, 766331573, 2193690811, 6289737611, 18081071971, 52099389811, 150344751721, 434277817873, 1255708046341, 3635404626381, 10538677900781, 30585912440557
OFFSET
0,5
COMMENTS
Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
Self-convolution yields A245930.
Limit a(n+1)/a(n) = 3.
LINKS
FORMULA
G.f.: 1 / sqrt( AGM((1-x)^2 + 4*x^2, (1-x)^2 - 4*x^2) ).
G.f.: 1 / sqrt( AGM((1-x)^2, sqrt((1-x)^4 - 16*x^4)) ).
a(n) ~ 3^(n+1) / (4*n*sqrt(Pi*log(n))) * (1 + (log(3) - 3*log(2) - gamma) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 - 3*gamma*log(3)/4 + 27*log(2)^2/8 - 9*log(3)*log(2)/4 + 3*log(3)^2/8 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...
where
1/A(x) = 1 - x - 2*x^4 - 6*x^5 - 12*x^6 - 20*x^7 - 42*x^8 - 126*x^9 - 392*x^10 - 1080*x^11 - 2722*x^12 - 6886*x^13 - 18612*x^14 - 52780*x^15 - 149564*x^16 -...
equals sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 + 2017*x^8 + 6721*x^9 + 21121*x^10 + 61065*x^11 + 171497*x^12 +...+ A245932(n)*x^n +...
MATHEMATICA
Simplify[CoefficientList[Series[Sqrt[(2*EllipticK[1 - (1 - 3*x)^4/(1 + x)^4]) / Pi] / (1 + x), {x, 0, 30}], x]] (* Vaclav Kotesovec, Sep 27 2019 *)
PROG
(PARI) {a(n)=local(A=1); A = 1 / sqrt( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
CROSSREFS
Cf. A245930, A245932 (log), A227846.
Sequence in context: A379463 A376835 A107587 * A190590 A341705 A241693
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 14 2014
STATUS
approved