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

1, 4, 40, 544, 8536, 145504, 2618176, 48943360, 941244376, 18502137184, 370091343040, 7508629231360, 154145664817600, 3196100636757760, 66834662101834240, 1407913577733228544, 29849617614785770456, 636440695668355742560, 13638210075999240396736, 293565508750164008207104, 6344596821114216520841536

Offset: 0

Author

Hugo Pfoertner, Aug 06 2017

Keywords

• nonn
• easy

Comments

Sequence epsilon in Almkvist, Straten, Zudilin article.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
• Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
• Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See epsilon p. 3.
• Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
• Armin Straub and Wadim Zudilin, Positivity of rational functions and their diagonals, J. Approx. Theory, 195:57-69, 2015. arXiv:1312.3732 [math.NT].
• Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
• Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
• Jeremy Tan, Simplifying a binomial sum for bridge deals with specific voids, Mathematics Stack Exchange, 2024.

Crossrefs

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.)

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.

Programs

Mathematica

Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 23 2017 *)
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];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 10 2022 *)

PARI

C=binomial; a(n) = sum (k=0, n, C(n,k)^2 * C(k+k,n)^2);

Formula

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.

a(n) = Sum_{k=ceiling(n/2)..n} binomial(n,k)^2*binomial(2*k,n)^2. [Gorodetsky] - Michel Marcus, Feb 25 2021

a(n) ~ 2^(2*n - 3/4) * (1 + sqrt(2))^(2*n+1) / (Pi*n)^(3/2). - Vaclav Kotesovec, Jul 10 2021

From Peter Bala, Apr 10 2022: (Start)

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)

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).

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)

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

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

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