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E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^4 = (2 - A(x)^4)^2.
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%I #20 Jun 06 2017 02:56:28

%S 1,1,-3,9,-87,705,-10395,144585,-2851695,56867265,-1413148275,

%T 36699287625,-1106370671175,35311847796225,-1256361047016075,

%U 47461118535455625,-1950838291460433375,84992806074321770625,-3968100259495356859875,195665521053499007135625

%N E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^4 = (2 - A(x)^4)^2.

%H Robert Israel, <a href="/A249789/b249789.txt">Table of n, a(n) for n = 0..384</a>

%F E.g.f.: (1 + 4*x*sqrt(1-4*x^2))^(1/4).

%F E.g.f.: (1 + 4*Series_Reversion(G(x)))^(1/4), where G(x) = ((1+4*x)^(1/2) - (1-4*x)^(1/2))/4.

%F E.g.f. A(x) satisfies:

%F (1) A(x)^4 + A(-x)^4 = 2.

%F (2) A(x)^2 - A(-x)^2 = 4*x.

%F (3) x = (A(x)^2 - (2 - A(x)^4)^(1/2))/4.

%F a(n) ~ (-1)^(n+1) * 2^(3*n/2-1/4) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 15 2014

%F a(n+4) = 3*(2*n+5)*(2*n+3)*a(n+2)-8*(2*n+3)*(2*n-1)*(n+2)*(n+1)*a(n). - _Robert Israel_, May 15 2017

%e E.g.f.: A(x) = 1 + x - 3*x^2/2! + 9*x^3/3! - 87*x^4/4! + 705*x^5/5! - 10395*x^6/6! + 144585*x^7/7! - 2851695*x^8/8! + 56867265*x^9/9! +...

%e Related series.

%e A(x)^2 = 1 + 2*x - 2*x^2 - 2*x^4 - 4*x^6 - 10*x^8 - 28*x^10 - 84*x^12 +...

%e A(x)^4 = 1 + 4*x - 8*x^3 - 8*x^5 - 16*x^7 - 40*x^9 - 112*x^11 - 336*x^13 +...

%p f:= gfun:-rectoproc({a(n+4) = (3*(2*n+5))*(2*n+3)*a(n+2)-(8*(2*n+3))*(2*n-1)*(n+2)*(n+1)*a(n), a(0)=1,a(1)=1,a(2)=-3,a(3)=9},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, May 15 2017

%t CoefficientList[Series[(1 + 4*x*Sqrt[1-4*x^2])^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Nov 15 2014 *)

%o (PARI) a(n)=local(X=x+x^2*O(x^n), A=(1 + 4*X*sqrt(1-4*X^2))^(1/4)); n!*polcoeff(A, n)

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A249786, A249788.

%K sign

%O 0,3

%A _Paul D. Hanna_, Nov 15 2014