%I M4020 #312 Aug 04 2024 13:29:09
%S 1,5,73,1445,33001,819005,21460825,584307365,16367912425,468690849005,
%T 13657436403073,403676083788125,12073365010564729,364713572395983725,
%U 11111571997143198073,341034504521827105445,10534522198396293262825,327259338516161442321485
%N Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
%C Conjecture: For each n = 1,2,3,... the Apéry polynomial A_n(x) = Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2*x^k is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 21 2013
%C The expansions of exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 5*x + 49*x^2 + 685*x^3 + 11807*x^4 + 232771*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + 3*x + 27*x^2 + 390*x^3 + 7038*x^4 + 144550*x^5 + ... both appear to have integer coefficients. See A267220. - _Peter Bala_, Jan 12 2016
%C Diagonal of the rational function R(x, y, z, w) = 1 / (1 - (w*x*y*z + w*x*y + w*z + x*y + x*z + y + z)); also diagonal of rational function H(x, y, z, w) = 1/(1 - w*(1+x)*(1+y)*(1+z)*(x*y*z + y*z + y + z + 1)). - _Gheorghe Coserea_, Jun 26 2018
%C Named after the French mathematician Roger Apéry (1916-1994). - _Amiram Eldar_, Jun 10 2021
%D Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.
%D Wolfram Koepf, Hypergeometric Identities. Ch. 2 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 55, 119 and 146, 1998.
%D Maxim Kontsevich and Don Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
%D Leonard Lipshitz and Alfred van der Poorten, "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990), pp. 339-358.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A005259/b005259.txt">Table of n, a(n) for n = 0..656</a> (first 101 terms from T. D. Noe)
%H Boris Adamczewski, Jason P. Bell and Eric Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.
%H Jean-Paul Allouche, <a href="http://www.math.jussieu.fr/~allouche/bibliorecente.html">A remark on Apéry's numbers</a>, J. Comput. Appl. Math., Vol. 83 (1997), pp. 123-125.
%H Roger Apéry, <a href="http://www.numdam.org/book-part/AST_1979__61__11_0/">Irrationalité de zeta(2) et zeta(3)</a>, in Journées 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. Astérisque, Vol. 61 (1979), pp. 11-13.
%H Roger Apéry, <a href="http://www.numdam.org/item?id=GAU_1981-1982__9_1_A9_0">Sur certaines séries entières arithmétiques</a>, Groupe de travail d'analyse ultramétrique, Vol. 9, No. 1 (1981-1982), Exp. No. 16, 2 p.
%H Roger Apéry, <a href="https://someclassicalmaths.files.wordpress.com/2011/10/apery-1981-paper.pdf">Interpolation de fractions continues et irrationalité de certaines constantes</a>, Bulletin de la section des sciences du C.T.H.S III (1981), pp. 37-53.
%H Thomas Baruchel and Carsten Elsner, <a href="http://arxiv.org/abs/1602.06445">On error sums formed by rational approximations with split denominators</a>, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
%H Frits Beukers, <a href="http://dx.doi.org/10.1016/0022-314X(87)90025-4">Another congruence for the Apéry numbers</a>, J. Number Theory, Vol. 25, No. 2 (1987), pp. 201-210.
%H Frits Beukers, <a href="https://www.researchgate.net/profile/F-Beukers/publication/27708775_Consequences_of_Apery%27s_work_on_z3">Consequences of Apéry's work on zeta(3)</a>, in "Zeta(3) irrationnel: les retombées", Rencontres Arithmétiques de Caen, June 2-3, 1995 [Mentions divisibility of a(n) by powers of 5 and powers of 11]
%H Francis Brown, <a href="http://arxiv.org/abs/1412.6508">Irrationality proofs for zeta values, moduli spaces and dinner parties</a>, arXiv:1412.6508 [math.NT], 2014.
%H William Y. C. Chen, Qing-Hu Hou and Yan-Ping Mu, <a href="http://dx.doi.org/10.1016/j.cam.2005.10.010">A telescoping method for double summations</a>, J. Comp. Appl. Math., Vol. 196, No. 2 (2006), pp. 553-566, Example 4.
%H Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
%H M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H Eric Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apéry-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
%H Emeric Deutsch and Bruce E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004.
%H Emeric Deutsch and Bruce E. Sagan, <a href="https://doi.org/10.1016/j.jnt.2005.06.005">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.
%H Gerald A. Edgar, <a href="https://web.archive.org/web/20150911173217/http://mathforum.org/kb/message.jspa?messageID=3704516&tstart=0">A formula with Legendre polynomials</a>, Sci. Math. Research posting Mar 21 2005.
%H G. A. Edgar, <a href="https://arxiv.org/abs/2005.10733">The Apéry Numbers as a Stieltjes Moment Sequence</a>, arXiv:2005.10733 [math.CA], 2020.
%H Carsten Elsner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., Vol. 43, No. 1 (2005), pp. 31-45.
%H Carsten Elsner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Elsner/elsner7.html">On prime-detecting sequences from Apéry's recurrence formulas for zeta(3) and zeta(2)</a>, JIS, Vol. 11 (2008), Article 08.5.1.
%H Stéphane Fischler, <a href="https://arxiv.org/abs/math/0303066">Irrationalité de valeurs de zeta</a>, arXiv:math/0303066 [math.NT], 2003.
%H Scott Garrabrant and Igor Pak, <a href="http://arxiv.org/abs/1407.8222">Counting with irrational tiles</a>, arXiv:1407.8222 [math.CO], 2014.
%H Ira Gessel, <a href="https://doi.org/10.1016/0022-314X(82)90071-3">Some congruences for Apéry numbers</a>, Journal of Number Theory, Vol. 14, No. 3 (1982) 362-368.
%H Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See gamma p. 3.
%H Richard K. Guy, <a href="/A005258/a005258.pdf">Letter to N. J. A. Sloane, Oct 1985</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apéry sequences with powers of binomial coefficients</a>, Nov 04 2012
%H Leonard Lipshitz and Alfred J. van der Poorten, <a href="https://doi.org/10.1515/9783110848632-029">Rational functions, diagonals, automata and arithmetic</a>, in: Richard A. Mollin (ed.), Number theory, Proceedings of the First Conference of the Canadian Number Theory Association Held at the Banff Center, Banff, Alberta, April 17-27, 1988, de Gruyter, 2016, pp. 339-358; <a href="https://citeseerx.ist.psu.edu/pdf/dfa9976c3141d5a40b1cf14231cbcbc85504b61e">alternative link</a>; <a href="https://web.archive.org/web/20040405052230/http://www-centre.mpce.mq.edu.au/alfpapers/a084.pdf">Wayback Machine copy</a>.
%H Ji-Cai Liu, <a href="https://arxiv.org/abs/1803.11442">Supercongruences for the (p-1)th Apéry number</a>, arXiv:1803.11442 [math.NT], 2018.
%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, Vol. 2 (2016), Article 5.
%H Stephen Melczer and Bruno Salvy, <a href="http://arxiv.org/abs/1605.00402">Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables</a>, arXiv:1605.00402 [cs.SC], 2016.
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H Robert Osburn and Brundaban Sahu, <a href="http://doi.org/10.7169/facm/2013.48.1.3">A supercongruence for generalized Domb numbers</a>, Funct. Approx. Comment. Math., Vol. 48, No. 1 (2013), pp. 29-36; <a href="https://maths.ucd.ie/~osburn/superdomb.pdf">alternative link</a>.
%H Math Overflow, <a href="http://mathoverflow.net/questions/178790/a-conjectured-formula-for-apery-numbers">A conjectured formula for Apéry numbers</a>
%H Eric Rowland and Reem Yassawi, <a href="http://www.numdam.org/item/JTNB_2015__27_1_245_0/">Automatic congruences for diagonals of rational functions</a>, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; <a href="http://arxiv.org/abs/1310.8635">arXiv preprint</a>, arXiv:1310.8635 [math.NT], 2013-2014.
%H Eric Rowland, Reem Yassawi and Christian Krattenthaler, <a href="https://arxiv.org/abs/2005.04801">Lucas congruences for the Apéry numbers modulo p^2</a>, arXiv:2005.04801 [math.NT], 2020.
%H Andrew Strangeway, <a href="http://arxiv.org/abs/1302.5089">A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space</a>, arXiv preprint arXiv:1302.5089 [math.AG], 2013.
%H Andrew Strangeway, <a href="http://dx.doi.org/10.1215/00277630-2817545">Quantum reconstruction for Fano bundles on projective space</a>, Nagoya Math. J., Vol. 218 (2015), pp. 1-28.
%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%H Volker Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
%H Volker Strehl, <a href="http://dx.doi.org/10.1016/0012-365X(94)00118-3">Binomial identities -- combinatorial and algorithmic aspects</a>, Discrete Mathematics, Vol. 136 (1994), 309-346.
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1112.1034">Congruences for Franel numbers</a>, arXiv preprint arXiv:1112.1034 [math.NT], 2011.
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2012.05.014">On sums of Apéry polynomials and related congruences</a>, J. Number Theory 132(2012), 2673-2699. [_Zhi-Wei Sun_, Mar 21 2013]
%H Zhi-Wei Sun. Sun, <a href="http://arxiv.org/abs/1101.1946">On sums of Apéry polynomials and related congruences</a>, arXiv:1101.1946 [math.NT], 2011-2014. [_Zhi-Wei Sun_, Mar 21 2013]
%H Alfred van der Poorten, <a href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf">A proof that Euler missed ...</a>, Math. Intelligencer, Vol. 1, No. 4 (December 1979), pp. 196-203, (b_n) after eq. (1.2), and Exercise 3.
%H Chen Wang, <a href="https://arxiv.org/abs/1909.08983">Two congruences concerning Apéry numbers</a>, arXiv:1909.08983 [math.NT], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AperyNumber.html">Apéry Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrehlIdentities.html">Strehl Identities</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.
%H Ernest X. W. Xia and Olivia X. M. Yao, <a href="https://doi.org/10.37236/3412">A Criterion for the Log-Convexity of Combinatorial Sequences</a>, The Electronic Journal of Combinatorics, Vol. 20 (2013), #P3.
%F D-finite with recurrence (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 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_ Jul 24 2002
%F a(n) = Sum_{k >= 0} A063007(n, k)*A000172(k)). A000172 = Franel numbers. - _Philippe Deléham_, Aug 14 2003
%F G.f.: (-1/2)*(3*x - 3 + (x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],[1],(-1/2)*(x^2 - 7*x + 1)*(x+1)^(-3)*(x^2 - 34*x + 1)^(1/2)+(1/2)*(x^3 + 30*x^2 - 24*x + 1)*(x+1)^(-3))^2. - _Mark van Hoeij_, Oct 29 2011
%F Let g(x, y) = 4*cos(2*x) + 8*sin(y)*cos(x) + 5 and let P(n,z) denote the Legendre polynomial of degree n. Then G. A. Edgar posted a conjecture of Alexandru Lupas that a(n) equals the double integral 1/(4*Pi^2)*int {y = -Pi..Pi} int {x = -Pi..Pi} P(n,g(x,y)) dx dy. (Added Jan 07 2015: Answered affirmatively in Math Overflow question 178790) - _Peter Bala_, Mar 04 2012; edited by _G. A. Edgar_, Dec 10 2016
%F a(n) ~ (1+sqrt(2))^(4*n+2)/(2^(9/4)*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Nov 01 2012
%F a(n) = Sum_{k=0..n} C(n,k)^2 * C(n+k,k)^2. - _Joerg Arndt_, May 11 2013
%F 0 = (-x^2+34*x^3-x^4)*y''' + (-3*x+153*x^2-6*x^3)*y'' + (-1+112*x-7*x^2)*y' + (5-x)*y, where y is g.f. - _Gheorghe Coserea_, Jul 14 2016
%F From _Peter Bala_, Jan 18 2020: (Start)
%F a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j) * C(n,k)^2 * C(n+k,k)^2 * C(n,j) * C(n+k+j,k+j).
%F a(n) = Sum_{0 <= j, k <= n} C(n,k) * C(n+k,k) * C(k,j)^3 (see Koepf, p. 55).
%F a(n) = Sum_{0 <= j, k <= n} C(n,k)^2 * C(n,j)^2 * C(3*n-j-k,2*n) (see Koepf, p. 119).
%F Diagonal coefficients of the rational function 1/((1 - x - y)*(1 - z - t) - x*y*z*t) (Straub, 2014). (End)
%F a(n) = [x^n] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^m at m = 2. At m = 1 we get the Apéry numbers A005258. - _Peter Bala_, Dec 22 2020
%F a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n, k). - _Peter Bala_, Jul 18 2024
%e G.f. = 1 + 5*x + 73*x^2 + 1445*x^3 + 33001*x^4 + 819005*x^5 + 21460825*x^6 + ...
%e a(2) = (binomial(2,0) * binomial(2+0,0))^2 + (binomial(2,1) * binomial(2+1,1))^2 + (binomial(2,2) * binomial(2+2,2))^2 = (1*1)^2 + (2*3)^2 + (1*6)^2 = 1 + 36 + 36 = 73. - _Michael B. Porter_, Jul 14 2016
%p 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;
%p # Alternative:
%p a := n -> hypergeom([-n, -n, 1+n, 1+n], [1, 1, 1], 1):
%p seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Jan 19 2020
%t Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1,1,1}, 1],{n,0,13}] (* _Jean-François Alcover_, Apr 01 2011 *)
%t Table[Sum[(Binomial[n,k]Binomial[n+k,k])^2,{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Oct 15 2011 *)
%t a[ n_] := SeriesCoefficient[ SeriesCoefficient[ SeriesCoefficient[ SeriesCoefficient[ 1 / (1 - t (1 + x ) (1 + y ) (1 + z ) (x y z + (y + 1) (z + 1))), {t, 0, n}], {x, 0, n}], {y, 0, n}], {z, 0, n}]; (* _Michael Somos_, May 14 2016 *)
%o (PARI) a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k,k))^2) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Haskell)
%o a005259 n = a005259_list !! n
%o a005259_list = 1 : 5 : zipWith div (zipWith (-)
%o (tail $ zipWith (*) a006221_list a005259_list)
%o (zipWith (*) (tail a000578_list) a005259_list)) (drop 2 a000578_list)
%o -- _Reinhard Zumkeller_, Mar 13 2014
%o (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k)^2)); # _Muniru A Asiru_, Sep 28 2018
%o (Magma) [&+[Binomial(n, k) ^2 *Binomial(n+k, k)^2: k in [0..n]]:n in [0..17]]; // _Marius A. Burtea_, Jan 20 2020
%o (Python)
%o def A005259(n):
%o m, g = 1, 0
%o for k in range(n+1):
%o g += m
%o m *= ((n+k+1)*(n-k))**2
%o m //=(k+1)**4
%o return g # _Chai Wah Wu_, Oct 02 2022
%Y Apéry's number or Apéry's constant zeta(3) is A002117. - _N. J. A. Sloane_, Jul 11 2023
%Y Cf. A002736, A005258, A005429, A005430, A059415, A059416, A063007, A000172.
%Y Cf. A006221, A000578, A006353.
%Y Related to diagonal of rational functions: A268545-A268555.
%Y 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.)
%Y 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.
%Y Cf. A092826 (prime terms).
%K nonn,easy,nice
%O 0,2
%A _Simon Plouffe_, _N. J. A. Sloane_, May 20 1991
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