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A104134 Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0). 9
1, -2, 40, -3680, 880000, -435776000, 386949376000, -560034421760000, 1233482823823360000, -3926150877331865600000, 17346066637844488192000000, -102987227337891283042304000000, 800183462504065339211776000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

cm(z):=sum((-1)^n*a(n)*z^(3*n)/(3*n)!,n=0..infinity) satisfies sm'(z)=cm(z)^2, cm'(z)=-sm(z)^2 with sm(0)=0 and cm(0)=1. Parametrizes Fermat's cubic X^3+Y^3=1.

Restated with different terminology: the functions sm(x,0) and cm(x,0) satisfy the following initial value problem: d(sm(x,0))/dx = (cm(x,0))^2; d(cm(x,0))/dx = - (sm(x,0))^2; sm(0,0) = 0; cm(0,0) = 1; The functions sm(x,0) and cm(x,0) are elliptic functions which satisfy the equation: (sm(x,0))^3 + (cm(x,0)^3) = 1.

The Dixonian elliptic function cm(z) parametrizes X^3+Y^3=1.

REFERENCES

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See p. 3.

A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy=1, Quarterly J. Pure Appl. Math. 24 (1890), 167-233.

E. van Fossen Conrad, Some continued fraction expansions of elliptic functions, PhD thesis, The Ohio State University, 2002.

LINKS

Table of n, a(n) for n=0..12.

R. Bacher and P. Flajolet, Pseudo-Factorials, Elliptic Functions and Continued Fractions, arXiv:0901.1379 [math.CA], 2009.

P. Flajolet, Publications

E. van Fossen Conrad and P. Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.

P. Lindqvist and J. Peetre, Two remarkable identities, called twos, for inverses to some Abelian integrals, Amer. Math. Monthly 108:5, 2001, 403-410.

E. Lundberg, On hypergoniometric functions of complex variables (at Jaak Peetre's home page)

FORMULA

G.f.: cm(u, 0).

E.g.f.: Sum_{k>=0} a(k) * x^(3*k) / (3*k)! = cm(x, 0). - Michael Somos, Aug 17 2007

EXAMPLE

cm(w) = 1 - (1/3)*w^3 + (1/18)*w^6 - (23/2268)*w^9 + (25/13608)*w^12 - ...

MAPLE

L:=proc(f) expand(x^2*diff(f, y)+y^2*diff(f, x)); end; Lit:=proc(f, m) if m=0 then f else L(Lit(f, m-1)) fi; end; seq(subs(x=0, y=1, Lit(y, 3*j)), j=0..20);

MATHEMATICA

nmax = 12; cm[z_] := (3*WeierstrassPPrime[z, {0, 1/27}] + 1) / (3*WeierstrassPPrime[z, {0, 1/27}] - 1); coes = CoefficientList[ Series[ cm[z], {z, 0, 3*nmax}], z][[1 ;; 3*nmax+1]]*Range[0, 3*nmax]!; a[n_] := coes[[3*n+1]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 04 2012 *)

a[ n_] := If[ n < 0, 0, With[ {m = 3 n}, m! SeriesCoefficient[ (3 WeierstrassPPrime[ x, {0, 1/27}] + 1) / (3 WeierstrassPPrime[ x, {0, 1/27}] - 1), {x, 0, m}]] ]; (* Michael Somos, Jun 09 2015 *)

m = 12; is = InverseSeries[Integrate[Normal[1/(1-x^3)^(2/3)+O[x]^(3m)], {x, 0, s}]+O[s]^(3m), s]; a[n_] := Coefficient[(1-is^3)^(1/3), s^(3n)]*(3n)!; a[0] = 1; Table[a[n], {n, 0, m}] (* Jean-François Alcover, Aug 30 2015 *)

PROG

(PARI) {a(n) = my(A, m); if( n<0, 0, A = O(x); for(i=0, n, A = 1 - intformal(intformal(A^2)^2) ); m = 3*n; m! * polcoeff( A, m))}; /* Michael Somos, Aug 17 2007 */

CROSSREFS

Cf. A104133.

Sequence in context: A209289 A246742 A293950 * A328553 A162868 A059476

Adjacent sequences:  A104131 A104132 A104133 * A104135 A104136 A104137

KEYWORD

sign

AUTHOR

Eric van Fossen Conrad (econrad(AT)math.ohio-state.edu), Mar 07 2005

EXTENSIONS

Additional comments and more terms from Philippe Flajolet, Jul 09 2005

Entry revised by N. J. A. Sloane, Dec 02 2005, Aug 17 2007

Signs added by Michael Somos, Aug 17 2007

STATUS

approved

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Last modified December 8 00:04 EST 2021. Contains 349590 sequences. (Running on oeis4.)