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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 (list; graph; refs; listen; history; text; internal format)
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

Paul D. Hanna, Table of n, a(n) for n = 0..1000

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: A097081 A093406 A107587 * A190590 A241693 A087323

Adjacent sequences:  A245928 A245929 A245930 * A245932 A245933 A245934

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 14 2014

STATUS

approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)