cp's OEIS Frontend

A125143 Almkvist-Zudilin numbers: Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3*k) * (3*k)!) / (k!)^3) * binomial(n,3*k) * binomial(n+k,k).

%I A125143 #109 Dec 24 2024 12:58:51
%S A125143 1,-3,9,-3,-279,2997,-19431,65853,292329,-7202523,69363009,-407637387,
%T A125143 702049401,17222388453,-261933431751,2181064727997,-10299472204311,
%U A125143 -15361051476987,900537860383569,-10586290198314843,74892552149042721,-235054958584593843
%N A125143 Almkvist-Zudilin numbers: Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3*k) * (3*k)!) / (k!)^3) * binomial(n,3*k) * binomial(n+k,k).
%C A125143 Apart from signs, this is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A125143 Diagonal of rational function 1/(1 - (x + y + z + w - 27*x*y*z*w)). - _Gheorghe Coserea_, Oct 14 2018
%C A125143 Named after the Swedish mathematician Gert Einar Torsten Almkvist (1934-2018) and the Russian mathematician Wadim Walentinowitsch Zudilin (b. 1970). - _Amiram Eldar_, Jun 23 2021
%D A125143 G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values. In Mirror Symmetry V, N. Yui, S.-T. Yau, and J. D. Lewis (eds.), AMS/IP Studies in Advanced Mathematics 38 (2007), International Press and Amer. Math. Soc., pp. 481-515. Cited in Chan & Verrill.
%D A125143 Helena Verrill, in a talk at the annual meeting of the Amer. Math. Soc., New Orleans, LA, Jan 2007 on "Series for 1/pi".
%H A125143 Seiichi Manyama, <a href="/A125143/b125143.txt">Table of n, a(n) for n = 0..1052</a> (terms 0..200 from Arkadiusz Wesolowski)
%H A125143 Gert Almkvist, Christian Krattenthaler, and Joakim Petersson, <a href="http://www.emis.de/journals/EM/expmath/volumes/12/12.4/Almkvist.pdf">Some new formulas for pi</a>, Experiment. Math., Vol. 12 (2003), pp. 441-456. (Math Rev MR2043994 by W. Zudilin)
%H A125143 G. Almkvist and W. Zudilin, <a href="https://arxiv.org/abs/math/0402386">Differential equations, mirror maps and zeta values</a>, arXiv:math/0402386 [math.NT], 2004.
%H A125143 Tewodros Amdeberhan, and Roberto Tauraso, <a href="http://arxiv.org/abs/1506.08437">Supercongruences for the Almkvist-Zudilin numbers</a>, arXiv:1506.08437 [math.NT], 2015.
%H A125143 Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle and Armin Straub, <a href="https://arxiv.org/abs/1804.10929">Diagonal asymptotics for symmetric rational functions via ACSV</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, arXiv:1804.10929 [math.CO], 2018.
%H A125143 Heng Huat Chan and Helena Verrill, <a href="http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0003/MRL-2009-0016-0003-a003.pdf">The Apery numbers, the Almkvist-Zudilin numbers and new series for 1/Pi</a>, Math. Res. Lett., Vol. 16, No. 3 (2009), pp. 405-420.
%H A125143 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. See Table 2 p. 7.
%H A125143 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 delta p. 3.
%H A125143 Ji-Cai Liu, <a href="https://arxiv.org/abs/2008.06675">A p-adic analogue of Chan and Verrill's formula for 1/Pi</a>, arXiv:2008.06675 [math.NT], 2020.
%H A125143 Ji-Cai Liu, <a href="https://ssmr.ro/bulletin/volumes/67-4/node9.html">A supercongruence related to Ramanujan-type formula for 1/Pi</a>, Bull. Math. Soc. Sci. Math. Roumanie Tome 67 (115), No. 4, 2024, 483-492. See p. 483.
%H A125143 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, Vol. 2, (2016), Article 5.
%H A125143 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A125143 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2002.12072">Super congruences concerning binomial coefficients and Apéry-like numbers</a>, arXiv:2002.12072 [math.NT], 2020.
%H A125143 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020.
%F A125143 a(n) = Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3*k) * (3*k)!) / (k!)^3) * binomial(n,3*k) * binomial(n+k,k) . - _Arkadiusz Wesolowski_, Jul 13 2011
%F A125143 Recurrence: n^3*a(n) = -(2*n-1)*(7*n^2 - 7*n + 3)*a(n-1) - 81*(n-1)^3*a(n-2). - _Vaclav Kotesovec_, Sep 11 2013
%F A125143 Lim sup n->infinity |a(n)|^(1/n) = 9. - _Vaclav Kotesovec_, Sep 11 2013
%F A125143 G.f. y=A(x) satisfies: 0 = x^2*(81*x^2 + 14*x + 1)*y''' + 3*x*(162*x^2 + 21*x + 1)*y'' + (21*x + 1)*(27*x + 1)*y' + 3*(27*x + 1)*y. - _Gheorghe Coserea_, Oct 15 2018
%F A125143 G.f.: hypergeom([1/8, 5/8], [1], -256*x^3/((81*x^2 + 14*x + 1)*(-x + 1)^2))^2/((81*x^2 + 14*x + 1)^(1/4)*sqrt(-x + 1)). - _Sergey Yurkevich_, Aug 31 2020
%t A125143 Table[Sum[(-1)^(n-k)*((3^(n-3*k)*(3*k)!)/(k!)^3) *Binomial[n,3*k] *Binomial[n+k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%o A125143 (PARI) a(n) = sum(k=0,n, (-1)^(n-k)*((3^(n-3*k)*(3*k)!)/(k!)^3)*binomial(n,3*k)*binomial(n+k,k) );
%Y A125143 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 A125143 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 A125143 easy,sign
%O A125143 0,2
%A A125143 _R. K. Guy_, Jan 11 2007
%E A125143 Edited and more terms added by _Arkadiusz Wesolowski_, Jul 13 2011