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A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.

%I A005259 M4020 #323 Mar 12 2025 13:17:23
%S A005259 1,5,73,1445,33001,819005,21460825,584307365,16367912425,468690849005,
%T A005259 13657436403073,403676083788125,12073365010564729,364713572395983725,
%U A005259 11111571997143198073,341034504521827105445,10534522198396293262825,327259338516161442321485
%N A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
%C A005259 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 A005259 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 A005259 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 A005259 Named after the French mathematician Roger Apéry (1916-1994). - _Amiram Eldar_, Jun 10 2021
%D A005259 Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.
%D A005259 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 A005259 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 A005259 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 A005259 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A005259 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 A005259 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 A005259 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 A005259 Gerald A. Edgar, <a href="https://web.archive.org/web/20150911173217/http://mathforum.org/kb/message.jspa?messageID=3704516&amp;tstart=0">A formula with Legendre polynomials</a>, Sci. Math. Research posting Mar 21 2005.
%H A005259 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 A005259 Carsten Elsner, <a href="https://web.archive.org/web/2024*/https://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 A005259 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 A005259 Stéphane Fischler, <a href="https://arxiv.org/abs/math/0303066">Irrationalité de valeurs de zeta</a>, arXiv:math/0303066 [math.NT], 2003.
%H A005259 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 A005259 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 A005259 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 A005259 Richard K. Guy, <a href="/A005258/a005258.pdf">Letter to N. J. A. Sloane, Oct 1985</a>
%H A005259 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 A005259 Antoine Labelle et al., <a href="https://mathoverflow.net/q/489184">A formula for Apéry numbers</a>, MathOverflow, 2025.
%H A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 Math Overflow, <a href="http://mathoverflow.net/questions/178790/a-conjectured-formula-for-apery-numbers">A conjectured formula for Apéry numbers</a>
%H A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A005259 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 A005259 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 A005259 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 A005259 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 A005259 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 A005259 Chen Wang, <a href="https://arxiv.org/abs/1909.08983">Two congruences concerning Apéry numbers</a>, arXiv:1909.08983 [math.NT], 2019.
%H A005259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AperyNumber.html">Apéry Number</a>.
%H A005259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StrehlIdentities.html">Strehl Identities</a>.
%H A005259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.
%H A005259 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 A005259 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 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_ Jul 24 2002
%F A005259 a(n) = Sum_{k >= 0} A063007(n, k)*A000172(k). A000172 = Franel numbers. - _Philippe Deléham_, Aug 14 2003
%F A005259 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 A005259 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 A005259 a(n) ~ (1+sqrt(2))^(4*n+2)/(2^(9/4)*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Nov 01 2012
%F A005259 a(n) = Sum_{k=0..n} C(n,k)^2 * C(n+k,k)^2. - _Joerg Arndt_, May 11 2013
%F A005259 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 A005259 From _Peter Bala_, Jan 18 2020: (Start)
%F A005259 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 A005259 a(n) = Sum_{0 <= j, k <= n} C(n,k) * C(n+k,k) * C(k,j)^3 (see Koepf, p. 55).
%F A005259 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 A005259 Diagonal coefficients of the rational function 1/((1 - x - y)*(1 - z - t) - x*y*z*t) (Straub, 2014). (End)
%F A005259 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 A005259 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n, k). - _Peter Bala_, Jul 18 2024
%F A005259 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,k)^2 * C(n,j)^2 * C(k+j,k), see Labelle et al. link. - _Max Alekseyev_, Mar 12 2025
%e A005259 G.f. = 1 + 5*x + 73*x^2 + 1445*x^3 + 33001*x^4 + 819005*x^5 + 21460825*x^6 + ...
%e A005259 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 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;
%p A005259 # Alternative:
%p A005259 a := n -> hypergeom([-n, -n, 1+n, 1+n], [1, 1, 1], 1):
%p A005259 seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Jan 19 2020
%t A005259 Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1,1,1}, 1],{n,0,13}] (* _Jean-François Alcover_, Apr 01 2011 *)
%t A005259 Table[Sum[(Binomial[n,k]Binomial[n+k,k])^2,{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Oct 15 2011 *)
%t A005259 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 A005259 (PARI) a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k,k))^2) \\ _Charles R Greathouse IV_, Nov 20 2012
%o A005259 (Haskell)
%o A005259 a005259 n = a005259_list !! n
%o A005259 a005259_list = 1 : 5 : zipWith div (zipWith (-)
%o A005259    (tail $ zipWith (*) a006221_list a005259_list)
%o A005259    (zipWith (*) (tail a000578_list) a005259_list)) (drop 2 a000578_list)
%o A005259 -- _Reinhard Zumkeller_, Mar 13 2014
%o A005259 (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 A005259 (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 A005259 (Python)
%o A005259 def A005259(n):
%o A005259     m, g = 1, 0
%o A005259     for k in range(n+1):
%o A005259         g += m
%o A005259         m *= ((n+k+1)*(n-k))**2
%o A005259         m //=(k+1)**4
%o A005259     return g # _Chai Wah Wu_, Oct 02 2022
%Y A005259 Apéry's number or Apéry's constant zeta(3) is A002117. - _N. J. A. Sloane_, Jul 11 2023
%Y A005259 Cf. A002736, A005258, A005429, A005430, A059415, A059416, A063007, A000172.
%Y A005259 Cf. A006221, A000578, A006353.
%Y A005259 Related to diagonal of rational functions: A268545-A268555.
%Y A005259 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 A005259 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 A005259 Cf. A092826 (prime terms).
%K A005259 nonn,easy,nice
%O A005259 0,2
%A A005259 _Simon Plouffe_, _N. J. A. Sloane_, May 20 1991