%I A005259 M4020
%S A005259 1,5,73,1445,33001,819005,21460825,584307365,16367912425,468690849005,
%T A005259 13657436403073,403676083788125,12073365010564729,364713572395983725
%N A005259 Apery (Ap\'{e}ry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
%C A005259 Prime Apery numbers include a(1) = 5, a(2) = 73, a(12) = 12073365010564729 
               and a(24). Semiprime central Delannoy numbers include a(4) = 33001 
               = 61 * 541. - Jonathan Vos Post (jvospost3(AT)gmail.com), May 22 
               2005
%D A005259 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005259 R. 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 A005259 R. Apery, ``Interpolation de fractions continues et irrationalit\'{e} 
               de certaines constantes,'' in Math\'{e}matiques, Minist\`{e}re universit\'{e}s 
               (France), Comit\'{e} travaux historiques et scientifiques. Bull. 
               Section Sciences, Vol. 3, pp. 243-246, 1981.
%D A005259 F. Beukers, Another congruence for the Apery numbers. J. Number Theory 
               25 (1987), no. 2, 201-210.
%D A005259 C. Elsner, On recurrence formulae for sums involving binomial coefficients, 
               Fib. Q., 43 (No. 1, 2005), 31-45.
%D A005259 W. Koepf, Hypergeometric Summation, Vieweg, 1998, p. 146.
%D A005259 M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and 
               W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., 
               Springer-Verlag, 2001.
%D A005259 Volker Strehl, Binomial identities -- combinatorial and algorithmic aspects, 
               Discrete Mathematics, Vol. 136 (1994), 309-346. [From David Callan 
               (callan(AT)stat.wisc.edu), Aug 27 2009]
%H A005259 T. D. Noe, <a href="b005259.txt">Table of n, a(n) for n=0..100</a>
%H A005259 J.-P. Allouche, <a href="http://www.lri.fr/~allouche/bibliorecente.html">
               A remark on Apery's numbers</a>, J. Comput. Appl. Math. 83 (1997), 
               123-125.
%H A005259 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 A005259 S. Fischler, <a href="http://arXiv.org/abs/math.NT/0303066">Irrationalit\'e 
               de valeurs de z\^eta</a>
%H A005259 L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/
               alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</
               a>
%H A005259 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">
               Recurrences and Legendre transform</a>
%H A005259 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 A005259 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StrehlIdentities.html">Strehl Identities</a>
%H A005259 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SchmidtsProblem.html">Schmidt's Problem</a>
%F A005259 (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 
               1.
%F A005259 Representation as a special value of the hypergeometric function 4F3, 
               in Maple notation: a(n)=hypergeom([n+1, n+1, -n, -n], [1, 1, 1], 
               1), n=0, 1... - Karol A. Penson (penson(AT)lptl.jussieu.fr) Jul 24 
               2002
%F A005259 a(n) = Sum( k>=0, A063007(n, k)*A000172(k)). A000172 = Franel numbers. 
               - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003
%p A005259 a := proc(n) option remember; if n=0 then 1 elif n=1 then 5 else (n^(-3))* 
               ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); 
               fi; end;
%Y A005259 Cf. A002736, A005258, A005259, A005429, A005430, A059415, A059416.
%Y A005259 Cf. A063007, A000172.
%Y A005259 Sequence in context: A155662 A159509 A127167 this_sequence A062440 A126748 
               A048144
%Y A005259 Adjacent sequences: A005256 A005257 A005258 this_sequence A005260 A005261 
               A005262
%K A005259 nonn,easy,nice
%O A005259 0,2
%A A005259 Simon Plouffe (simon.plouffe(AT)gmail.com), N. J. A. Sloane (njas(AT)research.att.com).