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A295870 a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient. 3
1, 12, 660, 48720, 4005540, 349260912, 31626298704, 2940502593600, 278788387440420, 26831860080682800, 2613367831568654160, 257012469788428710720, 25479526081439438845200, 2543092744417831625342400, 255292245777771431285140800, 25755871314484468746363582720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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)".

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.

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.

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?

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?

REFERENCES

D. Husemöller, Elliptic Curves, 2nd ed., New York: Springer, 2004.

J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer, 2009.

LINKS

Table of n, a(n) for n=0..15.

J. Cremona, Elliptic Curves over Q, LMFDB 2017.

B. Klee, The Virtues of X_{n+1} = (4+3*X_{n})/(3+2*X_{n}), seqfans mailing list, 2017.

B. Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.

Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018).

Bradley Klee, Phase Plane Geometry.

M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22.

P. Paule and M. Schorn, FastZeil: the Paule/Schorn implementation of Gosper's and Zeilberger's algorithm, RISC 2017; Local copy, pdf file only, no active links

D. Zeilberger, The Method of Creative Telescoping, Journal of Symbolic Computation, 11.3 (1991), 195-204.

FORMULA

a(n) = A005809(n)*A005721(n).

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).

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).

MATHEMATICA

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]}];

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}];

({#, SameQ[a/@Range[0, 15], #]}&@b[15])[[1]]

CROSSREFS

Cf. A000897, A002894, A006480, A113424.

Factors: A005190, A005809, A005721.

Complex Period: A300058.

Sequence in context: A195554 A220327 A281780 * A177322 A060612 A203307

Adjacent sequences:  A295867 A295868 A295869 * A295871 A295872 A295873

KEYWORD

nonn

AUTHOR

Bradley Klee, Feb 23 2018

STATUS

approved

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Last modified July 25 09:49 EDT 2021. Contains 346289 sequences. (Running on oeis4.)