A159601 - OEIS

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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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	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 - 2S(x)^2]^(1/4) converges to` `C(r) = gamma(3/4)^2/(Pi/2)^(3/2) = 0.7627597635...`
	LINKS	`Table of n, a(n) for n=1..14.`
	FORMULA	`E.g.f. S(x) satisfies: C(x)^4 + 2S(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 + 2S(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(2k+1)^2/(1 + 2x(k+1)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013`
	EXAMPLE	`E.g.f: S(x) = x - 3x^3/3! + 27x^5/5! - 441x^7/7! + 11529x^9/9! +...` `S(x)^2 = 2x^2/2! - 24x^4/4! + 504x^6/6! - 16128x^8/8! +-...` `C(x)^4 + 2S(x)^2 = 1 where:` `C(x) = 1 - x^2/2! + 3x^4/4! - 27x^6/6! + 441x^8/8! -+...` `C(x)^2 = 1 - 2x^2/2! + 12x^4/4! - 144x^6/6! + 3024x^8/8! -+...` `C(x)^3 = 1 - 3x^2/2! + 27x^4/4! - 441x^6/6! + 11529x^8/8! -+...` `C(x)^4 = 1 - 4x^2/2! + 48x^4/4! - 1008x^6/6! + 32256x^8/8! -+...` `1/C(x) = C(ix) = 1 + x^2/2! + 3x^4/4! + 27x^6/6! + 441x^8/8! +...` `log(C(x)) = -x^2/2! - 12x^6/6! - 3024x^10/10! - 4390848*x^14/14! -...` `Coefficients in log(C(x)) are given by A104203 (ignoring signs).`
	PROG	`(PARI) {a(n)=local(S=x); for(i=0, 2n, S=intformal((1-2S^2+O(x^(2n)))^(3/4))); (2n-1)!polcoeff(S, 2n-1)}`
	CROSSREFS	`Cf. A159600 (C(x)), A104203 (unsigned e.g.f. = S(x)/C(x)).` `Sequence in context: A136719 A279844 A159600 * A193541 A193544 A286306` `Adjacent sequences: A159598 A159599 A159600 * A159602 A159603 A159604`
	KEYWORD	`sign`
	AUTHOR	`Paul D. Hanna, May 08 2009`
	STATUS	`approved`

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)