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A001421 a(n) = (6*n)!/((n!)^3*(3*n)!).

%I A001421 #98 Jan 23 2025 00:30:43
%S A001421 1,120,83160,81681600,93699005400,117386113965120,155667030019300800,
%T A001421 214804163196079142400,305240072216678400087000,
%U A001421 443655767845074392936328000,656486312795713480715743268160,985646873056680684690542988249600,1497786250388951255453847206769124800
%N A001421 a(n) = (6*n)!/((n!)^3*(3*n)!).
%C A001421 Self-convolution of A092870, where A092870(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - _Paul D. Hanna_, Jan 25 2011
%H A001421 Seiichi Manyama, <a href="/A001421/b001421.txt">Table of n, a(n) for n = 0..310</a> (terms 0..75 from Vincenzo Librandi)
%H A001421 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A001421 M. Kaneko and D. Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998 (See Eq. 31.)
%H A001421 R. S. Maier, <a href="http://arxiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, arXiv:0807.1081 [math.NT], 2008_2010, p. 34 equation (7.29b).
%H A001421 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.
%F A001421 O.g.f.: Hypergeometric2F1(5/12, 1/12; 1; 1728x)^2. - Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009
%F A001421 a(n) = binomial(2n,n) * (12^n/n!^2) * Product_{k=0..n-1} (6k+1)*(6k+5). - _Paul D. Hanna_, Jan 25 2011
%F A001421 G.f.: F(1/6, 1/2, 5/6; 1, 1; 1728*x), a hypergeometric series. - _Michael Somos_, Feb 28 2011
%F A001421 0 = y^3*z^3 - 360*y^4*z^2 + 43200*y^5*z - 1728000*y^6 - 16632*x*y^2*z^3 + 7691328*x*y^3*z^2 - 1738520064*x*y^4*z + 176027074560*x*y^5 + 92207808*x^2*y*z^3 - 69176553984*x^2*y^2*z^2 + 23624298528768*x^2*y^3*z - 2853152143441920*x^2*y^4 - 170400029184*x^3*z^3 + 224945232150528*x^3*y*z^2 - 92759146352345088*x^3*y^2*z + 11686511179538104320*x^3*y^3 where x = a(n), y = a(n+1), z = a(n+2) for all n in z. - _Michael Somos_, Sep 21 2014
%F A001421 a(n) ~ 2^(6*n - 1) * 3^(3*n) / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Apr 07 2018
%F A001421 From _Peter Bala_, Feb 14 2020: (Start)
%F A001421 a(n) = binomial(6*n,n)*binomial(5*n,n)*binomial(4*n,n) = ( [x^n](1 + x)^(6*n) ) * ( [x^n](1 + x)^(5*n) ) * ( [x^n](1 + x)^(4*n) ) = [x^n](F(x)^(120*n)), where F(x) = 1 + x + 227*x^2 + 123980*x^3 + 92940839*x^4 + 82527556542*x^5 + 81459995686401*x^6 + ...
%F A001421 appears to have integer coefficients. For similar results see A008979.
%F A001421 a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
%F A001421 a(n) = [(x*y*z)^n] (1 + x + y + z)^(6*n). (End)
%F A001421 a(n) = (8^n/n!^3)*Product_{k = 0..3*n-1} (2*k + 1). - _Peter Bala_, Feb 26 2023
%F A001421 a(n) = 24*(6*n - 1)*(2*n - 1)*(6*n - 5)*a(n-1)/n^3. - _Neven Sajko_, Jul 19 2023
%F A001421 From _Karol A. Penson_, Jan 21 2025: (Start)
%F A001421 a(n) = Integral_{x=0..1728} x^n*W(x), with W(x) =  W1(x) + W2(x) + W3(x), where
%F A001421   W1(x) = hypergeometric([1/6, 1/6, 1/6], [1/3, 2/3], x/1728)/(6*sqrt(Pi)*x^(5/6)*Gamma(5/6)^3),
%F A001421   W2(x) = - hypergeometric([1/2, 1/2, 1/2], [2/3, 4/3], x/1728)/(24*Pi^2*sqrt(x)), and
%F A001421   W3(x) = hypergeometric([5/6, 5/6, 5/6], [4/3, 5/3], x/1728)*Gamma(5/6)^3/(1536*Pi^(7/2)*x^(1/6)). This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, 1728). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-1/6), and for x > 0 is monotonically decreasing to zero at x = 1728. (End)
%e A001421 G.f.: A(x) = 1 + 120*x + 83160*x^2 + 81681600*x^3 + ...
%e A001421 A(x)^(1/2) = 1 + 60*x + 39780*x^2 + 38454000*x^3 + ... + A092870(n)*x^n + ...
%p A001421 f := n->(6*n)!/( (n!)^3*(3*n)!);
%t A001421 Factorial[6 n]/(Factorial[3n] Factorial[n]^3) (* Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009 *)
%t A001421 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/6, 1/2, 5/6}, {1, 1}, 1728 x], {x, 0, n}] (* _Michael Somos_, Jul 11 2011 *)
%o A001421 (PARI) {a(n)=(2*n)!/n!^2*(12^n/n!^2)*prod(k=0, n-1, (6*k+1)*(6*k+5))} \\ _Paul D. Hanna_, Jan 25 2011
%o A001421 (Magma) [Factorial(6*n)/(Factorial(n)^3*Factorial(3*n)): n in [0..15]]; // _Vincenzo Librandi_, Oct 26 2011
%Y A001421 Cf. A092870; variants: A184423, A008977, A184892, A184896, A184898. - _Paul D. Hanna_, Jan 25 2011
%Y A001421 Cf. A289292.
%Y A001421 Cf. A145492, A145493, A145494, A008979.
%K A001421 nonn
%O A001421 0,2
%A A001421 _N. J. A. Sloane_, Glenn K Painter (KUPK78A(AT)prodigy.com)