A104133 Coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the Dixon elliptic function sm(x,0).

1, -4, 160, -20800, 6476800, -3946624000, 4161608704000, -6974121256960000, 17455222222028800000, -62226770432344883200000, 304379186781653598208000000, -1982049657077223312916480000000, 16758824127564135479341219840000000

Offset: 0

Author

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

Keywords

• sign

Comments

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

Some extracts from Kent E. Morrison's Math. Review (MR2223029) of the van Fossen Conrad - Flajolet paper: (Start)

Section 3 turns to the combinatorial significance of the integer coefficients of z^n/n! in the Taylor expansions of sm and cm. The numbers of certain urn histories for a particular Polya urn scheme have the signless versions of sm(z) and cm(z) as exponential generating functions. ...

Section 4 deals with a second combinatorial model. The same Dixonian functions appear as exponential generating functions of two classes of permutations. Permutations are represented by rooted trees, so that the level of a permutation element is defined. Also, each permutation element is one of the types peak, valley, double rise, or double fall according to its value relative to its two neighbors when the permutation is written as a word. The class of permutations whose elements at odd level are valleys only has exponential generating function -sm(-z), while the class whose elements at even level are valleys only has exponential generating function cm(-z). ...

In Section 5 the focus is on a third combinatorial model of r-repeated permutations. The Dixonian functions appear as generating functions of some 3-repeated permutations. ...

Section 6 gives some further examples of the appearance of the Dixonian functions in various contexts. ...

All in all this is a paper to recommend highly. (End)

Examples

			sm(w) = w - (1/6)*w^4 + (2/63)*w^7 - (13/2268)*w^10 + ...
		

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, Quart. 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. Minor changes.
• 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. A104134.

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(x,3*j+1)),j=0..20);

Mathematica

nmax = 12; sm[z_] := 6*WeierstrassP[z, {0, 1/27}] / (1 - 3*WeierstrassPPrime[z, {0, 1/27}]); coes = CoefficientList[ Series[ sm[z], {z, 0, 3*nmax+1}], z]*Range[0, 3*nmax+1]!; a[n_] := coes[[3*n+2]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 04 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, m! SeriesCoefficient[ 6 WeierstrassP[ x, {0, 1/27}] / (1 - 3 WeierstrassPPrime[ x, {0, 1/27}]), {x, 0, m}]] ]; (* Michael Somos, Jun 09 2015 *)
m = 12; is = InverseSeries[Integrate[Normal[1/(1-x^3)^(2/3)+O[x]^(3m+2)], {x, 0, s}]+O[s]^(3m+2), s]; a[n_] := Coefficient[is, s^(3n+1)]*(3n+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 = intformal( (1 - intformal(A^2))^2) ); m = 3*n + 1; m! * polcoeff( A, m))}; /* Michael Somos, Aug 17 2007 */

Formula

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

E.g.f.: Sum_{k>=0} a(k) * x^(3*k + 1) / (3*k + 1)! = sm(x, 0). - Michael Somos, Aug 17 2007

G.f.: 1/T(0), where T(k) = 1 + 2*x*(3*k+1)*((3*k+1)^2+1) - x^2*(3*k+1)*(3*k+2)^2*(3*k+3)^2*(3*k+4)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 01 2013

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