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

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

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). 1
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

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

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..200

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.

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

Sequence in context: A120429 A101431 A120982 * A200012 A130701 A202021

Adjacent sequences:  A125140 A125141 A125142 * A125144 A125145 A125146

KEYWORD

easy,sign

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 December 8 02:07 EST 2016. Contains 278902 sequences.