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A125143
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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).
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47
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1, -3, 9, -3, -279, 2997, -19431, 65853, 292329, -7202523, 69363009, -407637387, 702049401, 17222388453, -261933431751, 2181064727997, -10299472204311, -15361051476987, 900537860383569, -10586290198314843, 74892552149042721, -235054958584593843
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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".
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LINKS
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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)
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FORMULA
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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
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
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MATHEMATICA
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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 *)
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PROG
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(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) );
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CROSSREFS
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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.
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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