editing

approved

(* Calculation of constants {d, c}: *) {1/r, Sqrt[(s^2*(1 + (3 + 8*s)*(r + 4*r*s)^3 + r*(1 - 4*s - 16*s^2) - r^2*(5 + 40*s + 96*s^2 + 64*s^3))) / (4*Pi*(-1 + r)*(1 - 2*r*(1 + 14*s + 32*s^2) + r^2*(1 + 28*s + 192*s^2 + 384*s^3)))]} /. FindRoot[{Pi*s == 2*EllipticK[-16*r*s^2/((-1 + r)*(-1 + r*(1 + 4*s)^2))] / Sqrt[(-1 + r)*(-1 + r*(1 + 4*s)^2)], (-1 + r)*EllipticE[-16*r*s^2/((-1 + r)*(-1 + r*(1 + 4*s)^2))] / (s*(-1 + r + 4*r*s)*Sqrt[(-1 + r)*(-1 + r*(1 + 4*s)^2)]) == Pi}, {r, 1/60}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)

a(n) ~ c * d^n / n^(3/2), where d = 63.0960591646291970170956677412581268961854502622276974361888132988405... and c = 0.194671854097221132646990457090704594779965159082067267258127815911... - Vaclav Kotesovec, Jan 29 2024

approved

editing

Vaclav Kotesovec, <a href="/A247019/b247019_1.txt">Table of n, a(n) for n = 0..200</a>

editing

approved

Vaclav Kotesovec, <a href="/A247019/b247019_1.txt">Table of n, a(n) for n = 0..200</a>

approved

editing

editing

approved

G.f. satisfies: A(x) = Sum_{n>=0} x^n * [ Sum_{k=0..n} C(n,k) * C(2*k,k) * A(x)^k ]^2.

G.f. satisfies: A(x) = 1 / AGM(1 - x*(1+4*A(x)), sqrt((1-x)*(1-x*(1+4*A(x))^2)) ), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

G.f.: A(x) = 1 + 9*x + 229*x^2 + 8265*x^3 + 348349*x^4 + 16015309*x^5 +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 /agm(1-x*(1+4*A), sqrt((1-x)*(1-x*(1+4*A)^2 +x*O(x^n))) )); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 / agm(1-x*(1+4*A), sqrt((1-x)*(1-x*(1+4*A)^2 +x*O(x^n))) )); polcoeff(A, n)}

approved

editing

editing

approved

allocated for Paul D. Hanna

G.f. satisfies: A(x) = Sum_{n>=0} x^n * [ Sum_{k=0..n} C(n,k) * C(2*k,k) * A(x)^k ]^2.

1, 9, 229, 8265, 348349, 16015309, 778537149, 39348577689, 2046753733661, 108845415772629, 5890878814795401, 323408093688028781, 17966522918887184357, 1008117869403566779605, 57050765652157351532365, 3252460557853243186407529, 186619243293579737401397901, 10768643445899311909031558245

0,2

G.f. satisfies: A(x) = 1 / AGM(1 - x*(1+4*A(x)), sqrt((1-x)*(1-x*(1+4*A(x))^2)) ), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

G.f.: A(x) = 1 + 9*x + 229*x^2 + 8265*x^3 + 348349*x^4 + 16015309*x^5 +...

such that

A(x) = 1 + x*(1 + 1*2*A(x))^2 + x^2*(1 + 2*2*A(x) + 1*6*A(x)^2)^2

+ x^3*(1 + 3*2*A(x) + 3*6*A(x)^2 + 1*20*A(x)^3)^2

+ x^4*(1 + 4*2*A(x) + 6*6*A(x)^2 + 4*20*A(x)^3 + 1*70*A(x)^4)^2

+ x^5*(1 + 5*2*A(x) + 10*6*A(x)^2 + 10*20*A(x)^3 + 5*70*A(x)^4 + 1*252*A(x)^5)^2

+ x^6*(1 + 6*2*A(x) + 15*6*A(x)^2 + 20*20*A(x)^3 + 15*70*A(x)^4 + 6*252*A(x)^5 + 1*924*A(x)^6)^2 +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 /agm(1-x*(1+4*A), sqrt((1-x)*(1-x*(1+4*A)^2 +x*O(x^n))) )); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = sum(m=0, n, x^m*sum(k=0, m, binomial(m, k) * binomial(2*k, k) * A^k +x*O(x^n) )^2 ) ); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

allocated

nonn

Paul D. Hanna, Sep 09 2014

approved

editing