

A104133


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


11



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

0,2


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 rrepeated permutations. The Dixonian functions appear as generating functions of some 3repeated 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)


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), 167233.
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, P. Flajolet, Pseudofactorials, 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.
P. Lindqvist and J. Peetre, Two remarkable identities, called twos, for inverses to some Abelian integrals, Amer. Math. Monthly 108:5, 2001, 403410.
E. Lundberg, On hypergoniometric functions of complex variables (at Jaak Peetre's home page).


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


EXAMPLE

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


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, m1)) 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}] (* JeanFrançois 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/(1x^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}] (* JeanFrançois Alcover, Aug 30 2015 *)


PROG

(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 */


CROSSREFS

Cf. A104134.
Sequence in context: A224044 A224275 A224198 * A214007 A248330 A157463
Adjacent sequences: A104130 A104131 A104132 * A104134 A104135 A104136


KEYWORD

sign


AUTHOR

Eric van Fossen Conrad (econrad(AT)math.ohiostate.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



