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%I #106 Dec 26 2023 10:17:32
%S 1,60,13860,4084080,1338557220,465817912560,168470811709200,
%T 62588625639883200,23717177328413240100,9124964373613212524400,
%U 3553261127084984957001360,1397224499394244497967972800,553883078634868423069470550800,221068174083308549543680044926400
%N a(n) = (6*n)!/((3*n)!*(2*n)!*n!).
%C Appears in Ramanujan's theory of elliptic functions of signature 6.
%C 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
%H G. C. Greubel, <a href="/A113424/b113424.txt">Table of n, a(n) for n = 0..250</a>
%H J. Cremona, <a href="http://www.lmfdb.org/EllipticCurve/Q/">Elliptic Curves over Q</a>, LMFDB 2017.
%H Alin Bostan, Armin Straub, and Sergey Yurkevich, <a href="https://arxiv.org/abs/2212.10116">On the representability of sequences as constant terms</a>, arXiv:2212.10116 [math.NT], 2022.
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>
%H Bradley 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 Bradley Klee, <a href="https://groups.google.com/forum/#!topic/lmfdb-support/lSnR800nUac">On LMFDB period data</a>, LMFDB-support mailing list, 2018.
%H Bradley Klee, <a href="http://demonstrations.wolfram.com/WeierstrassSolutionOfCubicAnharmonicOscillation/"> Weierstrass Solution of Cubic Anharmonic Oscillation</a>, Wolfram Demonstrations Project, 2018.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf">Modular Equations and Approximations to Pi</a>, Quarterly Journal of Mathematics, XLV (1914), 350-372.
%H L. C. Shen, <a href="https://doi.org/10.1007/s11139-011-9360-8">A note on Ramanujan’s identities involving the hypergeometric function 2F1(1/6,5/6;1;z)</a>, The Ramanujan Journal, 30.2 (2013), 211-222.
%F G.f.: hypergeometric2F1(1/6, 5/6; 1; 432 * x).
%F a(n) ~ 432^n/(2*Pi*n). - _Ilya Gutkovskiy_, Oct 13 2016
%F a(n) = A005809(n)*A066802(n). - _Bradley Klee_, Feb 25 2018
%F 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
%F From _Peter Bala_, Feb 28 2020: (Start)
%F a(n) = C(6*n,2*n)*C(4*n,n).
%F a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k (apply Mestrovic, equation 39).
%F (-1)^n*a(n) = [x^(2*n)*y^(2*n)] ( (1 + x + y)*(1 - x + y) )^(4*n).
%F a(n) = [x^n] ( F(x) )^(60*n), where F(x) = 1 + x + 56*x^2 + 7355*x^3 + 1290319*x^4 + 264117464*x^5 + 59508459679*x^6 + ... appears to have integer coefficients. We conjecture that for k >= 1 the sequence defined by b_k(n) := [x^n] F(x)^(k*n) satisfies the above supercongruences for primes p >= 7. (End)
%F From _Peter Bala_, Mar 20 2022: (Start)
%F Right-hand side of the following identities valid for n >= 1:
%F Sum_{k = 0..2*n} 4*n*(4*n+k-1)!/(k!*n!*(3*n)!) = (6*n)!/((3*n)!*(2*n)!*n!);
%F Sum_{k = 0..3*n} 3*n*(3*n+k-1)!/(k!*n!*(2*n)!) = (6*n)!/((3*n)!(2*n)!*n!).
%F Cf. A001451. (End)
%F From _Peter Bala_, Feb 26 2023: (Start)
%F a(n) = (4^n/n!^2) * Product_{k = n..3*n-1} 2*k + 1.
%F a(n) = (12^n/n!^2) * Product_{k = 0..n-1} (6*k + 1)*(6*k + 5). (End)
%F a(n) = 12*(6*n - 1)*(6*n - 5)*a(n-1)/n^2. - _Neven Sajko_, Jul 19 2023
%F From _Karol A. Penson_, Dec 26 2023: (Start)
%F a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(18)*MeijerG([[], [0, 0]], [[-1/6, -5/6], []], x/432)/(1296*Pi), where MeijerG is the Meijer G - function.
%F Apparently, W(x) cannot be represented by any other function. W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.000368414. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem. (End)
%e G.f. = 1 + 60*x + 13860*x^2 + 4084080*x^3 + 1338557220*x^4 + ... - _Michael Somos_, Dec 02 2018
%t a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 5/6, 1, 432 x], {x, 0, n}];
%t Table[Multinomial[n, 2 n, 3 n], {n, 0, 15}] (* _Vladimir Reshetnikov_, Oct 12 2016 *)
%t a[ n_] := Multinomial[n, 2 n, 3 n]; (* _Michael Somos_, Dec 02 2018 *)
%o (PARI) {a(n) = if( n<0, 0, (6*n)! / ((3*n)! * (2*n)! * n!))};
%o (GAP) List([0..15],n->Factorial(6*n)/(Factorial(3*n)*Factorial(2*n)*Factorial(n))); # _Muniru A Asiru_, Apr 08 2018
%Y Elliptic Integrals: A002894, A006480, A000897. Factors: A005809, A066802.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Oct 31 2005
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