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A104134
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Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).
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2
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1, -2, 40, -3680, 880000, -435776000, 386949376000, -560034421760000, 1233482823823360000, -3926150877331865600000, 17346066637844488192000000, -102987227337891283042304000000, 800183462504065339211776000000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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REFERENCES
| R. BACHER AND P. FLAJOLET, PSEUDO-FACTORIALS, ELLIPTIC FUNCTIONS AND CONTINUED FRACTIONS, arXiv 0901.1379. [Added by N. J. A. Sloane (njas(AT)research.att.com), Feb 01 2009]
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.
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.
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LINKS
| P. Flajolet, Publications
E. Lundberg, On hypergoniometric functions of complex variables (at Jaak Peetre's home page)
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FORMULA
| G.f.: cm(u, 0).
G.f.: Sum_{k>=0} a(k) * x^(3*k)/(3*k)! = cm(x, 0). - Michael Somos, Aug 17 2007
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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);
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MATHEMATICA
| m = 13; a[0] = 1;
cm[z_] = Sum[a[n]*z^(3*n)/(3*n)!, {n, 0, m}] ;
sm[z_] = Normal[Series[Integrate[cm[z]^2, z], {z, 0, 3*m}]];
eq[0] = Thread[CoefficientList[sm[x]^3 + cm[x]^3 - 1 , x] == 0] // Union // Rest;
Do[s[k] = Solve[eq[k-1] // First, a[k]] // First;
eq[k] = eq[k-1] /. s[k] // Rest, {k, 1, m}]
Array[a, m, 0] /. Flatten[Array[s, m]]
(* From Jean-François Alcover, Jun 14 2011 *)
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PROG
| (PARI) {a(n) = local(A); if(n<0, 0, A = O(x); for(i=0, n, A = 1 - intformal(intformal(A^2)^2) ); n=3*n; n!*polcoeff( A, n))} - Michael Somos, Aug 17 2007
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CROSSREFS
| Cf. A104133.
Sequence in context: A000819 A060079 A052502 * A162868 A059476 A062769
Adjacent sequences: A104131 A104132 A104133 * A104135 A104136 A104137
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KEYWORD
| sign
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AUTHOR
| Eric van Fossen Conrad (econrad(AT)math.ohio-state.edu), Mar 07 2005
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EXTENSIONS
| Additional comments and more terms from Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Jul 09 2005
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 02 2005, Aug 17 2007
Signs added by Michael Somos, Aug 17 2007
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