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%I #49 Jan 02 2023 12:30:54
%S 1,12,660,48720,4005540,349260912,31626298704,2940502593600,
%T 278788387440420,26831860080682800,2613367831568654160,
%U 257012469788428710720,25479526081439438845200,2543092744417831625342400,255292245777771431285140800,25755871314484468746363582720
%N a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient.
%C Compare with EllipticK A002894 and the Ramanujan period-energy functions A113424, A006480, A000897. The series expansion "T(x) = 2*Pi*Sum_{n>=0} a_n*x^n" determines the real period T of elliptic curves in the family "x=p^2+q^2-4*(q^2-p^2)*q, 0 < x < 1/108". This sequence serves as a counterexample to the naive idea that elliptic integrals will always evaluate to a hypergeometric function such as 2F1(a,b;c;x).
%C A300058 is the complex period-energy function, after scaling energy and time dimensions such that all a(n) are integers and a(0)=1. The Picard-Fuchs equation is "(12-288*x+9216*x^2)*T(x) + (-1+232*x-8160*x^2+82944*x^3)*T'(x) + (-x+164*x^2-6432*x^3+41472*x^4)*T''(x)".
%C Although the sequence is not generated by a hypergeometric function, it can be formulated in terms of Hypergeometric numbers, specifically the binomial coefficients. Then Zeilberger's algorithm outputs a second order recurrence with polynomial coefficients.
%C The contour plot is nice to look at, with reflection symmetry, three critical points, and two separatrices dividing the phase plane into eight distinct regions.
%C Hyperbolic Critical points are located at (q,p) locations (1/6,0) and (-1/4,sqrt(5)/4) and (-1/4,-sqrt(5)/4). Is it possible to use chord-and-tangent addition rules to produce an exponentially-convergent Diophantine approximation to sqrt(5) that moves along the upper separatrix x=1/8?
%C Does there exist a period-preserving transformation that takes any one of the curves with 0 < x < 1/108 into a particular Weierstrass curve from the L-function and Modular Forms Database?
%D D. Husemöller, Elliptic Curves, 2nd ed., New York: Springer, 2004.
%D J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer, 2009.
%H J. Cremona, <a href="http://www.lmfdb.org/EllipticCurve/Q/">Elliptic Curves over Q</a>, LMFDB 2017.
%H B. Klee, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-September/017940.html">The Virtues of X_{n+1} = (4+3*X_{n})/(3+2*X_{n})</a>, seqfans mailing list, 2017.
%H B. Klee, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-December/018186.html">Geometric G.F. for Ramanujan Periods</a>, seqfans mailing list, 2017.
%H Brad Klee, <a href="http://demonstrations.wolfram.com/DerivingHypergeometricPicardFuchsEquations/">Deriving Hypergeometric Picard-Fuchs Equations</a>, Wolfram Demonstrations Project (2018).
%H Bradley Klee, <a href="/A295870/a295870.pdf">Phase Plane Geometry</a>.
%H M. Kontsevich and D. Zagier, <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf">Periods</a>, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22.
%H P. Paule and M. Schorn, <a href="http://www.risc.jku.at/research/combinat/software/ergosum/RISC/fastZeil.html">FastZeil: the Paule/Schorn implementation of Gosper's and Zeilberger's algorithm</a>, RISC 2017; <a href="/A295870/a295870_1.pdf">Local copy, pdf file only, no active links</a>
%H D. Zeilberger, <a href="https://doi.org/10.1016/S0747-7171(08)80044-2">The Method of Creative Telescoping</a>, Journal of Symbolic Computation, 11.3 (1991), 195-204.
%F a(n) = A005809(n)*A005721(n).
%F a(n) = Sum_{k=0..floor(3n/4)} ((-1)^k)*binomial(3*n,n)*binomial(2 *n, k)*binomial(5*n - 4*k - 1, 3*n - 4*k).
%F c1 = 8 *(-30 + 201*n - 319*n^2 + 145*n^3); c2 = -8640*(n - 5/3)*(n - 4/3)*(n - 1/5); c3 = 10*(n - 6/5)*n^2; a(0)=1; a(1)=12; a(n) = (c1/c3)*a(n-1) + (c2/c3)*a(n-2).
%t b[NN_]:=Total/@Table[((-1)^k)*Binomial[3*n,n]*Binomial[2*n,k]*Binomial[5*n-4*k-1,3*n-4*k],{n,0,NN},{k,0,Floor[3*n/4]}];
%t c1=8*(-30+201*n-319*n^2+145*n^3);c2=-8640*(n-5/3)*(n-4/3)*(n-1/5);c3=10*(n-6/5)*n^2;a[0]=1;a[1]=12;a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2],{n->n0}];
%t ({#,SameQ[a/@Range[0, 15],#]}&@b[15])[[1]]
%Y Cf. A000897, A002894, A006480, A113424.
%Y Factors: A005190, A005809, A005721.
%Y Complex Period: A300058.
%K nonn
%O 0,2
%A _Bradley Klee_, Feb 23 2018
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