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%I #73 Apr 03 2024 03:08:45
%S 1,4,40,544,8536,145504,2618176,48943360,941244376,18502137184,
%T 370091343040,7508629231360,154145664817600,3196100636757760,
%U 66834662101834240,1407913577733228544,29849617614785770456,636440695668355742560,13638210075999240396736,293565508750164008207104,6344596821114216520841536
%N Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.
%C Sequence epsilon in Almkvist, Straten, Zudilin article.
%H Seiichi Manyama, <a href="/A290575/b290575.txt">Table of n, a(n) for n = 0..500</a>
%H G. Almkvist, D. van Straten, and W. Zudilin, <a href="https://doi.org/10.1017/S0013091509000959">Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations</a>, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
%H Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023.
%H Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See epsilon p. 3.
%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5
%H Armin Straub and Wadim Zudilin, <a href="https://arxiv.org/abs/1312.3732">Positivity of rational functions and their diagonals</a>, J. Approx. Theory, 195:57-69, 2015. arXiv:1312.3732 [math.NT].
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%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.
%H Jeremy Tan, <a href="https://math.stackexchange.com/a/4888024/357390">Simplifying a binomial sum for bridge deals with specific voids</a>, Mathematics Stack Exchange, 2024.
%F a(-1)=0, a(0)=1, a(n+1) = ((2*n+1)*(12*n^2+12*n+4)*a(n)-16*n^3*a(n-1))/(n+1)^3.
%F a(n) = Sum_{k=ceiling(n/2)..n} binomial(n,k)^2*binomial(2*k,n)^2. [Gorodetsky] - _Michel Marcus_, Feb 25 2021
%F a(n) ~ 2^(2*n - 3/4) * (1 + sqrt(2))^(2*n+1) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Jul 10 2021
%F From _Peter Bala_, Apr 10 2022: (Start)
%F The g.f. is the diagonal of the rational function 1/(1 - (x + y + z + t) + 2*(x*y*z + x*y*t + x*z*t + y*z*t) + 4*x*y*z*t) (Straub and Zudilin)
%F The g.f. appears to be the diagonal of the rational function 1/(1 - x - y + z - t - 2*(x*z + y*z + z*t) + 4*(x*y*t + x*z*t) + 8*x*y*z*t).
%F If true, then a(n) = [(x*y*z)^n] ( (x + y + z + 1)*(x + y + z - 1)*(x + y - z - 1)*(x - y - z + 1) )^n . (End)
%F a(n) = binomial(2*n, n)^2 * hypergeom([1/2-n/2, 1/2-n/2, -n/2, -n/2], [1, 1/2-n, 1/2-n], 1). - _Peter Luschny_, Apr 10 2022
%F G.f.: hypergeom([1/8, 3/8],[1], 256*x^2 / (1 - 4*x)^4)^2 / (1 - 4*x). - _Mark van Hoeij_, Nov 12 2022
%F a(n) = [(w*x*y*z)^n] ((w+z)*(x+z)*(y+z)*(w+x+y+z))^n = Sum_{0 <= j <= i <= n} binomial(n,i)^2*binomial(i,j)^2*binomial(n+j,i). - _Jeremy Tan_, Mar 28 2024
%t Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* _G. C. Greubel_, Oct 23 2017 *)
%t a[n_] := Binomial[2 n, n]^2 HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1/2 - n, 1/2 - n}, 1];
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Apr 10 2022 *)
%o (PARI) C=binomial; a(n) = sum (k=0, n, C(n,k)^2 * C(k+k,n)^2);
%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.)
%Y For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
%K nonn,easy
%O 0,2
%A _Hugo Pfoertner_, Aug 06 2017
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