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A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2. 46
1, 4, 40, 544, 8536, 145504, 2618176, 48943360, 941244376, 18502137184, 370091343040, 7508629231360, 154145664817600, 3196100636757760, 66834662101834240, 1407913577733228544, 29849617614785770456, 636440695668355742560, 13638210075999240396736, 293565508750164008207104, 6344596821114216520841536 (list; graph; refs; listen; history; text; internal format)



Sequence epsilon in Almkvist, Straten, Zudilin article.


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.

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

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.


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


Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 23 2017 *)


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


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.)

Other Apery-like sequences are A000172, A002893, A002895, A005258, A005259, A005260, A006077, A081085, A093388, A125143, A183204, A219692, A229111, A290576.

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.

Sequence in context: A074637 A075878 A092812 * A196867 A276362 A302178

Adjacent sequences:  A290572 A290573 A290574 * A290576 A290577 A290578




Hugo Pfoertner, Aug 06 2017



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Last modified February 28 14:16 EST 2021. Contains 341707 sequences. (Running on oeis4.)