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%I #89 Mar 25 2024 21:25:31
%S 1,8,88,1088,14296,195008,2728384,38879744,561787864,8206324928,
%T 120929313088,1794924383744,26802975999424,402298219288064,
%U 6064992788397568,91786654611673088,1393772628452578264,21227503080738294464,324160111169327247424
%N G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.
%D M. Petkovsek et al., "A=B", Peters, p. ix of second printing.
%H Reinhard Zumkeller, <a href="/A036917/b036917.txt">Table of n, a(n) for n = 0..500</a>
%H B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.
%H E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apery-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
%H 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 Ji-Cai Liu and He-Xia Ni, <a href="https://arxiv.org/abs/2004.07652">Supercongruences for Almkvist--Zudilin sequences</a>, arXiv:2004.07652 [math.NT], 2020. See Vn.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020. See Vn.
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/2005.02081">Congruences for two types of Apery-like sequences</a>, arXiv:2005.02081 [math.NT], 2020.
%F a(n) = (16*(n-1/2)*(2*n^2-2*n+1)*a(n-1)-256*(n-1)^3*a(n-2))/n^3.
%F a(n) = Sum_{k=0..n} (C(2 * (n-k), n-k) * C(2 * k, k))^2. [corrected by _Tito Piezas III_, Oct 19 2010]
%F a(n) = hypergeom([1/2, 1/2, -n, -n], [1, 1/2-n, 1/2-n], 1) * 4^n * (2n-1)!!^2 / n!^2. - _Vladimir Reshetnikov_, Mar 08 2014
%F a(n) ~ 2^(4*n+1) * log(n) / (n*Pi^2) * (1 + (4*log(2) + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Nov 28 2015
%F G.f. y=A(x) satisfies: 0 = x^2*(16*x - 1)^2*y''' + 3*x*(16*x - 1)*(32*x - 1)*y'' + (1792*x^2 - 112*x + 1)*y' + 8*(32*x - 1)*y. - _Gheorghe Coserea_, Jul 03 2018
%F G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - _Vaclav Kotesovec_, Oct 01 2019
%F It appears that a(n) is equal to the coefficient of (x*y*z*t)^n in the expansion of (1+x+y+z-t)^n * (1+x+y-z+t)^n * (1+x-y+z+t)^n * (1-x+y+z+t)^n. Cf. A000172. - _Peter Bala_, Sep 21 2021
%F G.f. y = A(x) satisfies 0 = x*(1 - 16*x)*(2*y''*y - y'*y') + 2*(1 - 32*x)*y*y' - 16*y*y. - _Michael Somos_, May 29 2023
%F Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - _Michael Somos_, May 30 2023
%F From _Paul D. Hanna_, Mar 25 2024: (Start)
%F G.f. ( Sum_{n>=0} binomial(2*n,n)^2 * x^n )^2.
%F G.f. Sum_{n>=0} binomial(2*n,n)^3 * x^n * (1 - 16*x)^n. (End)
%e G.f. = 1 + 8*x + 88*x^2 + 1088*x^3 + 14296*x^5 + 195008*x^5 + ... - _Michael Somos_, May 29 2023
%t a[n_] := (16 (n - 1/2)(2*n^2 - 2*n + 1)a[n - 1] - 256(n - 1)^3 a[n - 2])/n^3; a[0] = 1; a[1] = 8; Array[a, 19, 0] (* Or *)
%t f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *)
%t lmt = 20; Take[ 4^Range[0, 2 lmt]*CoefficientList[ Series[(4/Pi^2) EllipticK[4 x^(1/2)]^2, {x, 0, lmt}], x^(1/2)], lmt] (* _Robert G. Wilson v_ *)
%t a[n_] := HypergeometricPFQ[{1/2, 1/2, -n, -n}, {1, 1/2-n, 1/2-n}, 1] * 4^n * (2n-1)!!^2 / n!^2 (* _Vladimir Reshetnikov_, Mar 08 2014 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* _Michael Somos_, May 30 2023 *)
%o (Haskell)
%o a036917 n = sum $ map
%o (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n]
%o -- _Reinhard Zumkeller_, May 24 2012
%o (PARI) for(n=0,25, print1(sum(k=0,n, (binomial(2*n-2*k,n-k) *binomial(2*k,k))^2), ", ")) \\ _G. C. Greubel_, Oct 24 2017
%o (PARI) {a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)}); /* _Michael Somos_, May 29 2023 */
%Y Cf. A002894, A036915, A057703.
%Y Cf. A007318, A036916, A036829.
%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Replaced complicated definition via a formula with simple generating function provided by _Vladeta Jovovic_, Dec 01 2003. Thanks to _Paul D. Hanna_ for suggesting this. - _N. J. A. Sloane_, Mar 25 2024
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