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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

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 Mathe-matics 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)

Tewodros Amdeberhan, Roberto Tauraso, Supercongruences for the Almkvist-Zudilin numbers, arXiv:1506.08437 [math.NT], 2015.

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.

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

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: A120429 A101431 A120982 * 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

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Last modified September 20 01:09 EDT 2017. Contains 292251 sequences.