login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 46
1, -3, 9, -3, -279, 2997, -19431, 65853, 292329, -7202523, 69363009, -407637387, 702049401, 17222388453, -261933431751, 2181064727997, -10299472204311, -15361051476987, 900537860383569, -10586290198314843, 74892552149042721, -235054958584593843 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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., 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. 12 (2003), 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, Roberto Tauraso, Supercongruences for the Almkvist-Zudilin numbers, arXiv:1506.08437 [math.NT], 2015.

Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle, 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. 16 (2009), no. 3, 405-420.

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.

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

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 *)

PROG

(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) );

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.

Sequence in context: A101431 A120982 A293634 * A200012 A130701 A202021

Adjacent sequences:  A125140 A125141 A125142 * A125144 A125145 A125146

KEYWORD

easy,sign,changed

AUTHOR

R. K. Guy, Jan 11 2007

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 03:36 EDT 2018. Contains 316330 sequences. (Running on oeis4.)