OFFSET
0,2
COMMENTS
Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 4*x * AGM(A(x)^2, A(x)^4).
(2) A(x) = 1 + 4*x * AGM(A(x)^3, (A(x)^2 + A(x)^4)/2).
(3) A(x) = 1 + 4 * Series_Reversion( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ).
(4) A( x/AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.
a(n) ~ c * d^n / n^(3/2), where d = 28.0338265004083388867842940412535265992903265132288705384671366058202... and c = 0.21370406929731394715730174119301970236922500578435406822814969355660... - Vaclav Kotesovec, Jan 29 2024
EXAMPLE
G.f.: A(X) = 1 + 4*x + 48*x^2 + 784*x^3 + 14784*x^4 + 302960*x^5 + 6554624*x^6 + 147336384*x^7 + 3407207936*x^8 + 80538522544*x^9 + 1937217000576*x^10 + ...
RELATED SERIES.
x / AGM((1 + 4*x)^2, (1 + 4*x)^4) = x - 12*x^2 + 92*x^3 - 576*x^4 + 3220*x^5 - 16784*x^6 + 83536*x^7 - 402560*x^8 + 1894308*x^9 - 8751600*x^10 + ...
where A( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.
A(x)^2 = 1 + 8*x + 112*x^2 + 1952*x^3 + 38144*x^4 + 799456*x^5 + 17566848*x^6 + 399375232*x^7 + 9315958784*x^8 + 221714573152*x^9 + ...
A(x)^3 = 1 + 12*x + 192*x^2 + 3568*x^3 + 72384*x^4 + 1554768*x^5 + 34760064*x^6 + 800484672*x^7 + 18858757632*x^8 + 452388579088*x^9 + ...
A(x)^4 = 1 + 16*x + 288*x^2 + 5696*x^3 + 120064*x^4 + 2646464*x^5 + 60279552*x^6 + 1407812352*x^7 + 33532936192*x^8 + 811514412736*x^9 + ...
(A(x)^2 + A(x)^4)/2 = 1 + 12*x + 200*x^2 + 3824*x^3 + 79104*x^4 + 1722960*x^5 + 38923200*x^6 + 903593792*x^7 + 21424447488*x^8 + 516614492944*x^9 + ...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, s*(s - 1) * Sqrt[(1 + s + s^2 + s^3)/(2*Pi*(4 + s + 2*s^2 + 2*s^3 - 14*s^4 + 9*s^5))]} /. FindRoot[{1 + 2*Pi*r*s^4 / EllipticK[1 - 1/s^4] == s, 2*Pi*r*(1 - 2*s^4) + (-1 + s) * EllipticE[1 - 1/s^4] + (-1 + s^4)*Pi*r*s/(-1 + s) == 0}, {r, 1/30}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
PROG
(PARI) /* From definition: A(x) = 1 + 4*x*AGM(A(x)^2, A(x)^4) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, A^4)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^3, (A(x)^2 + A(x)^4)/2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^3, (A^2 + A^4)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM((1+4*x)^2, (1+4*x)^4)) */
{a(n) = my(A=1); A = 1 + 4*serreverse(x/agm((1+4*x)^2, (1+4*x)^4 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 28 2024
STATUS
approved