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A113424
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a(n) = (6n)!/((3n)!(2n)!n!).
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9
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1, 60, 13860, 4084080, 1338557220, 465817912560, 168470811709200, 62588625639883200, 23717177328413240100, 9124964373613212524400, 3553261127084984957001360, 1397224499394244497967972800, 553883078634868423069470550800, 221068174083308549543680044926400
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OFFSET
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0,2
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COMMENTS
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Appears in Ramanujan's theory of elliptic functions of signature 6.
The family of elliptic curves "x=2*H=p^2+q^2-q^3, 0<x<4/27" generates these a_n as the coefficients of the period-energy function "T(x)=2*Pi*2F1(1/6,5/6;1;(27/4)*x)". Set y=(27/4)*x, the Weierstrass parameters of this family are g2=(1/12), g3=(1/216)(1-2*y), j=432/(y-y^2). Our current statistical estimates suggest that about 500000 of Q-curves in LMFDB belong to this family. - Bradley Klee, Feb 25 2018
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..250
J. Cremona, Elliptic Curves over Q, LMFDB 2017.
H. J. Brothers, Pascal's Prism: Supplementary Material
B. Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
B. Klee, On LMFDB period data , LMFDB-support mailing list, 2018.
B. Klee, Weierstrass Solution of Cubic Anharmonic Oscillation, Wolfram Demonstrations Project, 2018.
S. Ramanujan, Modular Equations and Approximations to Pi, Quarterly Journal of Mathematics, XLV (1914), 350-372.
L. C. Shen, A note on Ramanujan’s identities involving the hypergeometric function 2F1(1/6,5/6;1;z), The Ramanujan Journal, 30.2 (2013), 211-222.
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FORMULA
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G.f.: hypergeometric2F1(1/6, 5/6; 1; 432 * x).
a(n) ~ 432^n/(2*Pi*n). - Ilya Gutkovskiy, Oct 13 2016
a(n) = A005809(n)*A066802(n). - Bradley Klee, Feb 25 2018
0 = a(n)*(-267483013447680*a(n+2) +25577192448000*a(n+3) -204669037440*a(n+4) +372142500*a(n+5)) +a(n+1)*(+408751349760*a(n+2) -57870650880*a(n+3) +546809652*a(n+4) -1088188*a(n+5)) +a(n+2)*(-17884800*a(n+2) +21466920*a(n+3) - 295844*a(n+4) +693*a(n+5)) for all n in Z. - Michael Somos, May 16 2018
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EXAMPLE
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G.f. = 1 + 60*x + 13860*x^2 + 4084080*x^3 + 1338557220*x^4 + ... - Michael Somos, Dec 02 2018
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 5/6, 1, 432 x], {x, 0, n}];
Table[Multinomial[n, 2 n, 3 n], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 12 2016 *)
a[ n_] := Multinomial[n, 2 n, 3 n]; (* Michael Somos, Dec 02 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, (6*n)! / ((3*n)! * (2*n)! * n!))};
(GAP) List([0..15], n->Factorial(6*n)/(Factorial(3*n)*Factorial(2*n)*Factorial(n))); # Muniru A Asiru, Apr 08 2018
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CROSSREFS
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Elliptic Integrals: A002894, A006480, A000897. Factors: A005809, A066802.
Sequence in context: A248708 A184890 A295598 * A009564 A269762 A291912
Adjacent sequences: A113421 A113422 A113423 * A113425 A113426 A113427
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Oct 31 2005
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STATUS
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approved
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