A104134 Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).

1, -2, 40, -3680, 880000, -435776000, 386949376000, -560034421760000, 1233482823823360000, -3926150877331865600000, 17346066637844488192000000, -102987227337891283042304000000, 800183462504065339211776000000000

Offset: 0

Author

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

Keywords

• sign

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.

Examples

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

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

• 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.
• Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, The Wallis Products for Fermat Curves, Vietnam J. Math. (2023).
• 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)

Crossrefs

Cf. A104133.

Programs

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 *)

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 */

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

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