Search: id:A005258
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%I A005258 M3057
%S A005258 1,3,19,147,1251,11253,104959,1004307,9793891,96918753,970336269,
%T A005258 9807518757,99912156111,1024622952993,10567623342519,109527728400147,
%U A005258 1140076177397091,11911997404064793,124879633548031009
%N A005258 Apery numbers: Sum C(n,k)^2 * C(n+k,k), k=0..n.
%C A005258 Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion 
               of a special point on a curve described by Beauville.
%C A005258 Equals the main diagonal of square array A108625. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jun 14 2005
%D A005258 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005258 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.
%D A005258 Beukers, F.; Another congruence for the Apery numbers. J. Number Theory 
               25 (1987), no. 2, 201-210.
%D A005258 Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre 
               fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, 
               May 24 1982.
%D A005258 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], 
               Master's Thesis (unpublished), Aug 26 1983.
%D A005258 C. Elsner, On recurrence formulae for sums involving binomial coefficients, 
               Fib. Q., 43 (No. 1, 2005), 31-45.
%H A005258 Simon Plouffe, <a href="b005258.txt">Table of n, a(n) for n = 0..954</
               a>
%H A005258 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
               Congruences for Catalan and Motzkin numbers and related sequences</
               a>, J. Num. Theory 117 (2006), 191-215.
%H A005258 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/OEIS/b005258.txt">
               The first 2553 Apery numbers</a>
%H A005258 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">
               Recurrences and Legendre transform</a>
%H A005258 A. van der Poorten, <a href="http://www.ift.uni.wroc.pl/~mwolf/Poorten_MI_195_0.pdf"> 
               A proof that Euler missed ... Apery's proof of the irrationality 
               of zeta(3). An informal report.</a> Math. Intelligencer 1 (1978/79), 
               no 4, 195-203.
%H A005258 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AperyNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A005258 W. Zudilin, <a href="http://arXiv.org/abs/math.NT/0409023">Approximations 
               to -, di and trilogarithms</a>
%F A005258 a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 24 2003
%F A005258 (n+1)^2 a_{n+1} = (11n^2+11n+3) a_n+n^2 a_{n-1}. - Matthijs Coster, Apr 
               28, 2004
%F A005258 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
%p A005258 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
%Y A005258 Cf. A002736, A005258, A005259, A005429, A005430.
%Y A005258 Cf. A108625.
%Y A005258 Cf. A143413.
%Y A005258 Sequence in context: A095002 A080833 A073516 this_sequence A131551 A074546 
               A054316
%Y A005258 Adjacent sequences: A005255 A005256 A005257 this_sequence A005259 A005260 
               A005261
%K A005258 nonn,easy,nice
%O A005258 0,2
%A A005258 N. J. A. Sloane (njas(AT)research.att.com).
%E A005258 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

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