A369537 - OEIS

%I #12 Feb 17 2024 03:20:06

%S 1,4,48,784,14784,302960,6554624,147336384,3407207936,80538522544,

%T 1937217000576,47262640993344,1166745699940352,29090562313367104,

%U 731508300407392256,18530124876627212032,472416442490053386240,12112314681652019632304,312110730162591314249088

%N Expansion of g.f. A(x) satisfying A(x) = 1 + 4*x * AGM(A(x)^2, A(x)^4).

%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

%H Paul D. Hanna, <a href="/A369537/b369537.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = 1 + 4*x * AGM(A(x)^2, A(x)^4).

%F (2) A(x) = 1 + 4*x * AGM(A(x)^3, (A(x)^2 + A(x)^4)/2).

%F (3) A(x) = 1 + 4 * Series_Reversion( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ).

%F (4) A( x/AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.

%F a(n) ~ c * d^n / n^(3/2), where d = 28.0338265004083388867842940412535265992903265132288705384671366058202... and c = 0.21370406929731394715730174119301970236922500578435406822814969355660... - _Vaclav Kotesovec_, Jan 29 2024

%e 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 + ...

%e RELATED SERIES.

%e 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 + ...

%e where A( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e (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 + ...

%t (* 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 *)

%o (PARI) /* From definition: A(x) = 1 + 4*x*AGM(A(x)^2, A(x)^4) */

%o {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)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^3, (A(x)^2 + A(x)^4)/2) */

%o {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)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM((1+4*x)^2, (1+4*x)^4)) */

%o {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)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A171454, A272823, A369536, A369538, A369539.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 28 2024