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A104133
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Coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the Dixon elliptic function sm(x,0).
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12
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1, -4, 160, -20800, 6476800, -3946624000, 4161608704000, -6974121256960000, 17455222222028800000, -62226770432344883200000, 304379186781653598208000000, -1982049657077223312916480000000, 16758824127564135479341219840000000
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OFFSET
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0,2
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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sm(w) = w - (1/6)*w^4 + (2/63)*w^7 - (13/2268)*w^10 + ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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Eric van Fossen Conrad (econrad(AT)math.ohio-state.edu), Mar 07 2005
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EXTENSIONS
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STATUS
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approved
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