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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). 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

Almkvist, Gert; Krattenthaler, Christian; and Petersson, Joakim; Some new formulas for pi. Experiment. Math. 12 (2003), 441-456. (Math Rev MR2043994 by W. Zudilin)

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.

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

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

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Last modified April 25 18:03 EDT 2015. Contains 257076 sequences.