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).

1, -3, 9, -3, -279, 2997, -19431, 65853, 292329, -7202523, 69363009, -407637387, 702049401, 17222388453, -261933431751, 2181064727997, -10299472204311, -15361051476987, 900537860383569, -10586290198314843, 74892552149042721, -235054958584593843

Offset: 0

Author

R. K. Guy, Jan 11 2007

Keywords

• easy
• sign

Comments

Apart from signs, this is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

Diagonal of rational function 1/(1 - (x + y + z + w - 27*x*y*z*w)). - Gheorghe Coserea, Oct 14 2018

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

References

• 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.
• Helena Verrill, in a talk at the annual meeting of the Amer. Math. Soc., New Orleans, LA, Jan 2007 on "Series for 1/pi".

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1052 (terms 0..200 from Arkadiusz Wesolowski)
• Gert Almkvist, Christian Krattenthaler, and Joakim Petersson, Some new formulas for pi, Experiment. Math., Vol. 12 (2003), pp. 441-456. (Math Rev MR2043994 by W. Zudilin)
• G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values, arXiv:math/0402386 [math.NT], 2004.
• Tewodros Amdeberhan, and Roberto Tauraso, Supercongruences for the Almkvist-Zudilin numbers, arXiv:1506.08437 [math.NT], 2015.
• Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle and Armin Straub, Diagonal asymptotics for symmetric rational functions via ACSV, LIPIcs Proceedings of Analysis of Algorithms 2018, arXiv:1804.10929 [math.CO], 2018.
• Heng Huat Chan and Helena Verrill, The Apery numbers, the Almkvist-Zudilin numbers and new series for 1/Pi, Math. Res. Lett., Vol. 16, No. 3 (2009), pp. 405-420.
• Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
• Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See delta p. 3.
• Ji-Cai Liu, A p-adic analogue of Chan and Verrill's formula for 1/Pi, arXiv:2008.06675 [math.NT], 2020.
• Ji-Cai Liu, A supercongruence related to Ramanujan-type formula for 1/Pi, Bull. Math. Soc. Sci. Math. Roumanie Tome 67 (115), No. 4, 2024, 483-492. See p. 483.
• Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, Vol. 2, (2016), Article 5.
• Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
• Zhi-Hong Sun, Super congruences concerning binomial coefficients and Apéry-like numbers, arXiv:2002.12072 [math.NT], 2020.
• Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.

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[(-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 *)

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) );

Formula

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

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

Lim sup n->infinity |a(n)|^(1/n) = 9. - Vaclav Kotesovec, Sep 11 2013

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

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

Extensions

Edited and more terms added by Arkadiusz Wesolowski, Jul 13 2011