A249789 E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^4 = (2 - A(x)^4)^2.

1, 1, -3, 9, -87, 705, -10395, 144585, -2851695, 56867265, -1413148275, 36699287625, -1106370671175, 35311847796225, -1256361047016075, 47461118535455625, -1950838291460433375, 84992806074321770625, -3968100259495356859875, 195665521053499007135625

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..384

Formula

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

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.

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

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

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

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

a(n) ~ (-1)^(n+1) * 2^(3*n/2-1/4) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 15 2014

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

Example

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! +...
Related series.
A(x)^2 = 1 + 2*x - 2*x^2 - 2*x^4 - 4*x^6 - 10*x^8 - 28*x^10 - 84*x^12 +...
A(x)^4 = 1 + 4*x - 8*x^3 - 8*x^5 - 16*x^7 - 40*x^9 - 112*x^11 - 336*x^13 +...

Maple

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):
map(f, [$0..30]); # Robert Israel, May 15 2017

Mathematica

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

Prog

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

Crossrefs

Cf. A249786, A249788.

Sequence in context: A326159 A203559 A184098 * A067210 A018654 A003225

Adjacent sequences: A249786 A249787 A249788 * A249790 A249791 A249792

Keyword

sign

Author

Paul D. Hanna, Nov 15 2014

Status

approved