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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)
 OFFSET 0,2 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, W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295. 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. 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. MATHEMATICA Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 23 2017 *) PROG (PARI) C=binomial; a(n) = sum (k=0, n, C(n, k)^2 * C(k+k, n)^2); 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.) 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 KEYWORD nonn,easy AUTHOR Hugo Pfoertner, Aug 06 2017 STATUS approved

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Last modified October 26 02:11 EDT 2020. Contains 338026 sequences. (Running on oeis4.)