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A005258
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Apery numbers: Sum C(n,k)^2 * C(n+k,k), k=0..n.
(Formerly M3057)
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19
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1, 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.
Equals the main diagonal of square array A108625. - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 14 2005
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REFERENCES
| Roger Apery, Irrationalite de zeta(2) et zeta(3), in Journees Arith. de Luminy. Colloque International du Centre National de la Recherche Scientifique (CNRS) held at the Centre Universitaire de Luminy, Luminy, Jun 20-24, 1978. Asterisque, 61 (1979), 11-13.
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.
Beukers, F.; Another congruence for the Apery numbers. J. Number Theory 25 (1987), no. 2, 201-210.
Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Simon Plouffe, Table of n, a(n) for n = 0..954
Lalit Jain and Pavlos Tzermias, Beukers' integrals and Apery's recurrences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.1.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
S. Plouffe, The first 2553 Apery numbers
V. Strehl, Recurrences and Legendre transform
A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report. Math. Intelligencer 1 (1978/79), no 4, 195-203.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
W. Zudilin, Approximations to -, di and trilogarithms
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FORMULA
| a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 24 2003
(n+1)^2 a_{n+1} = (11n^2+11n+3) a_n+n^2 a_{n-1}. - Matthijs Coster, Apr 28, 2004
Let b(n) be the solution to the above recurrence with b(0) = 0, b(1) = 5. Then the b(n) are rational numbers with b(n)/a(n) -> zeta(2) very rapidly. The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^(n-1)*5/n^2 leads to a series acceleration formula: zeta(2) = 5 * sum {n = 1..inf} 1/(n^2*a(n)*a(n-1)) = 5*[1/(1*3) + 1/(2^2*3*19) + 1/(3^2*19*147) + ...]. Similar results hold for the constant e: see A143413. - Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008
G.f.: hypergeom([1/12, 5/12],[1],-1728*x^5*(1-11*x-x^2)/(1-12*x+14*x^2+12*x^3+x^4)^3) / (1-12*x+14*x^2+12*x^3+x^4)^(1/4) - Mark van Hoeij, Oct 25 2011.
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MAPLE
| seq(add((multinomial(n+k, n-k, k, k))*binomial(n, k), k=0..n), n=0..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 18 2006
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MATHEMATICA
| a[n_] := HypergeometricPFQ[ {n+1, -n, -n}, {1, 1}, 1]; Table[ a[n], {n, 0, 18}] (* From Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)
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CROSSREFS
| Cf. A002736, A005258, A005259, A005429, A005430, A108625, A143413.
Sequence in context: A095002 A080833 A073516 * A131551 A074546 A054316
Adjacent sequences: A005255 A005256 A005257 * A005259 A005260 A005261
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
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