A159601 E.g.f. S(x) satisfies: S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0.

1, -3, 27, -441, 11529, -442827, 23444883, -1636819569, 145703137041, -16106380394643, 2164638920874507, -347592265948756521, 65724760945840254489, -14454276753061349098587

Offset: 1

Author

Paul D. Hanna, May 08 2009

Keywords

• sign

Comments

E.g.f. S(x) is an odd function; zero terms are omitted.

Apart from signs and initial term, same as A159600.

Radius of convergence of S(x) is |x| < r where:

r = (1/2)*Pi^(3/2)/gamma(3/4)^2 ;

r = L/sqrt(2) where L=Lemniscate constant ;

r = 1.8540746773013719184338503471952600...

Although S(x) diverges at |x|=r, the power series expansion:

C(x) = [1 - 2*S(x)^2]^(1/4) converges to

C(r) = gamma(3/4)^2/(Pi/2)^(3/2) = 0.7627597635...

Examples

			E.g.f: S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...
S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
C(x)^4 + 2*S(x)^2 = 1 where:
C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+...
C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+...
C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+...
C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+...
1/C(x) = C(i*x) = 1 + x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! +...
log(C(x)) = -x^2/2! - 12*x^6/6! - 3024*x^10/10! - 4390848*x^14/14! -...
Coefficients in log(C(x)) are given by A104203 (ignoring signs).
		

Crossrefs

Cf. A159600 (C(x)), A104203 (unsigned e.g.f. = S(x)/C(x)).

Programs

PARI

{a(n)=local(S=x);for(i=0,2*n,S=intformal((1-2*S^2+O(x^(2*n)))^(3/4)));(2*n-1)!*polcoeff(S,2*n-1)}

Formula

E.g.f. S(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where

S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

E.g.f. S(x) satisfies: S(x)/C(x) = e.g.f. of unsigned A104203 where C(x)^4 + 2*S(x)^2 = 1.

a(n) = -A159600(n), n>0. - M. F. Hasler, Aug 31 2012

G.f.: (1- 1/Q(0))/x, where Q(k) = 1 + x*(2*k+1)^2/(1 + 2*x*(k+1)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013