This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036917 #93 Aug 07 2024 01:19:21
%S A036917 1,8,88,1088,14296,195008,2728384,38879744,561787864,8206324928,
%T A036917 120929313088,1794924383744,26802975999424,402298219288064,
%U A036917 6064992788397568,91786654611673088,1393772628452578264,21227503080738294464,324160111169327247424
%N A036917 G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.
%D A036917 M. Petkovsek et al., "A=B", Peters, p. ix of second printing.
%H A036917 Reinhard Zumkeller, <a href="/A036917/b036917.txt">Table of n, a(n) for n = 0..500</a>
%H A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 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 A036917 G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - _Vaclav Kotesovec_, Oct 01 2019
%F A036917 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 A036917 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 A036917 Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - _Michael Somos_, May 30 2023
%F A036917 From _Paul D. Hanna_, Mar 25 2024: (Start)
%F A036917 G.f. ( Sum_{n>=0} binomial(2*n,n)^2 * x^n )^2.
%F A036917 G.f. Sum_{n>=0} binomial(2*n,n)^3 * x^n * (1 - 16*x)^n. (End)
%e A036917 G.f. = 1 + 8*x + 88*x^2 + 1088*x^3 + 14296*x^5 + 195008*x^5 + ... - _Michael Somos_, May 29 2023
%t A036917 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 A036917 f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *)
%t A036917 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 A036917 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 A036917 a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* _Michael Somos_, May 30 2023 *)
%o A036917 (Haskell)
%o A036917 a036917 n = sum $ map
%o A036917 (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n]
%o A036917 -- _Reinhard Zumkeller_, May 24 2012
%o A036917 (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 A036917 (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 A036917 Cf. A002894, A036915, A057703.
%Y A036917 Cf. A007318, A036916, A036829.
%Y A036917 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 A036917 nonn,easy
%O A036917 0,2
%A A036917 _N. J. A. Sloane_
%E A036917 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