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A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.

%I A290575 #73 Apr 03 2024 03:08:45
%S A290575 1,4,40,544,8536,145504,2618176,48943360,941244376,18502137184,
%T A290575 370091343040,7508629231360,154145664817600,3196100636757760,
%U A290575 66834662101834240,1407913577733228544,29849617614785770456,636440695668355742560,13638210075999240396736,293565508750164008207104,6344596821114216520841536
%N A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.
%C A290575 Sequence epsilon in Almkvist, Straten, Zudilin article.
%H A290575 Seiichi Manyama, <a href="/A290575/b290575.txt">Table of n, a(n) for n = 0..500</a>
%H A290575 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 A290575 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 A290575 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 A290575 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 A290575 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 A290575 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A290575 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 A290575 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 A290575 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 A290575 a(n) = Sum_{k=ceiling(n/2)..n} binomial(n,k)^2*binomial(2*k,n)^2. [Gorodetsky] - _Michel Marcus_, Feb 25 2021
%F A290575 a(n) ~ 2^(2*n - 3/4) * (1 + sqrt(2))^(2*n+1) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Jul 10 2021
%F A290575 From _Peter Bala_, Apr 10 2022: (Start)
%F A290575 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 A290575 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 A290575 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 A290575 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 A290575 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 A290575 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 A290575 Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* _G. C. Greubel_, Oct 23 2017 *)
%t A290575 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 A290575 Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Apr 10 2022 *)
%o A290575 (PARI) C=binomial; a(n) = sum (k=0, n, C(n,k)^2 * C(k+k,n)^2);
%Y A290575 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 A290575 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 A290575 nonn,easy
%O A290575 0,2
%A A290575 _Hugo Pfoertner_, Aug 06 2017