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A005258 Apéry numbers: a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k).

%I A005258 M3057 #282 Apr 02 2025 05:14:43
%S A005258 1,3,19,147,1251,11253,104959,1004307,9793891,96918753,970336269,
%T A005258 9807518757,99912156111,1024622952993,10567623342519,109527728400147,
%U A005258 1140076177397091,11911997404064793,124879633548031009,1313106114867738897,13844511065506477501
%N A005258 Apéry numbers: a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k).
%C A005258 This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004
%C A005258 Equals the main diagonal of square array A108625. - _Paul D. Hanna_, Jun 14 2005
%C A005258 This sequence is t_5 in Cooper's paper. - _Jason Kimberley_, Nov 25 2012
%C A005258 Conjecture: For each n=1,2,3,... the polynomial a_n(x) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)*x^k is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 21 2013
%C A005258 Diagonal of rational functions 1/(1 - x - x*y - y*z - x*z - x*y*z), 1/(1 + y + z + x*y + y*z + x*z + x*y*z), 1/(1 - x - y - z + x*y + x*y*z), 1/(1 - x - y - z + y*z + x*z - x*y*z). - _Gheorghe Coserea_, Jul 07 2018
%D A005258 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%D A005258 S. Melczer, An Invitation to Analytic Combinatorics, 2021; p. 129.
%D A005258 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005258 Simon Plouffe, <a href="/A005258/b005258.txt">Table of n, a(n) for n = 0..954</a>
%H A005258 B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="https://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 A005258 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, 61 (1979), 11-13.
%H A005258 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, 9 no. 1 (1981-1982), Exp. No. 16, 2 p.
%H A005258 Thomas Baruchel and C. Elsner, <a href="https://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 A005258 Arnaud Beauville, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5543443c/f31.item">Les familles stables de courbes sur P_1 admettant quatre fibres singulières</a>, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, page 657.
%H A005258 F. Beukers, <a href="http://dx.doi.org/10.1016/0022-314X(87)90025-4">Another congruence for the Apéry numbers</a>, J. Number Theory 25 (1987), no. 2, 201-210.
%H A005258 A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
%H A005258 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 A005258 Shaun Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J. (2012).
%H A005258 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.
%H A005258 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A005258 E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apéry-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
%H A005258 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/abs/math.CO/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Number Theory 117 (2006), 191-215.
%H A005258 C. 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., 43,1 (2005), 31-45.
%H A005258 C. 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 11 (2008) 08.5.1.
%H A005258 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 D p. 2.
%H A005258 R. K. Guy, <a href="/A005258/a005258.pdf">Letter to N. J. A. Sloane, Oct 1985</a>
%H A005258 S. Herfurtner, <a href="https://doi.org/10.1007/BF01445211">Elliptic surfaces with four singular fibres</a>, Mathematische Annalen, 1991. <a href="https://archive.mpim-bonn.mpg.de/id/eprint/860/">Preprint</a>.
%H A005258 Michael D. Hirschhorn, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/53-1/HirschhornConnection5272014.pdf">A Connection Between Pi and Phi</a>, Fibonacci Quart. 53 (2015), no. 1, 42-47.
%H A005258 Lalit Jain and Pavlos Tzermias, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Tzermias/tzermias5.html">Beukers' integrals and Apéry's recurrences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.1.
%H A005258 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H A005258 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 A005258 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 A005258 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, 2016, 2:5.
%H A005258 R. Mestrovic, <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 A005258 Peter Paule and Carsten Schneider, <a href="https://doi.org/10.1016/S0196-8858(03)00016-2">Computer proofs of a new family of harmonic number identities</a>, Advances in Applied Mathematics (31), 359-378, (2003).
%H A005258 Simon Plouffe, <a href="http://plouffe.fr/OEIS/b005258.txt">The first 2553 Apéry numbers</a>
%H A005258 E. Rowland and R. Yassawi, <a href="http://arxiv.org/abs/1310.8635">Automatic congruences for diagonals of rational functions</a>, arXiv preprint arXiv:1310.8635 [math.NT], 2013.
%H A005258 V. 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 A005258 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A005258 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 A005258 A. van der Poorten, <a href="http://www.ift.uni.wroc.pl/~mwolf/Poorten_MI_195_0.pdf"> A proof that Euler missed ... Apéry'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="https://mathworld.wolfram.com/AperyNumber.html">Apéry Number.</a>
%H A005258 D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apéry-like recurrence equations</a>. See line D in sporadic solutions table of page 5.
%H A005258 W. Zudilin, <a href="http://arxiv.org/abs/math/0409023">Approximations to -, di- and tri-logarithms</a>, arXiv:math/0409023 [math.CA], 2004-2005.
%F A005258 a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - _Vladeta Jovovic_, Apr 24 2003
%F A005258 D-finite with recurrence: (n+1)^2 * a(n+1) = (11*n^2+11*n+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} 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_, Aug 14 2008
%F A005258 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
%F A005258 a(n) ~ ((11+5*sqrt(5))/2)^(n+1/2)/(2*Pi*5^(1/4)*n). - _Vaclav Kotesovec_, Oct 05 2012
%F A005258 1/Pi = 5*(sqrt(47)/7614)*Sum_{n>=0} (-1)^n a(n)*binomial(2n,n)*(682n+71)/15228^n. [Cooper, equation (4)] - _Jason Kimberley_, Nov 26 2012
%F A005258 a(-1 - n) = (-1)^n * a(n) if n>=0. a(-1 - n) = -(-1)^n * a(n) if n<0. - _Michael Somos_, Sep 18 2013
%F A005258 0 = a(n)*(a(n+1)*(+4*a(n+2) + 83*a(n+3) - 12*a(n+4)) + a(n+2)*(+32*a(n+2) + 902*a(n+3) - 147*a(n+4)) + a(n+3)*(-56*a(n+3) + 12*a(n+4))) + a(n+1)*(a(n+1)*(+17*a(n+2) + 374*a(n+3) - 56*a(n+4)) + a(n+2)*(+176*a(n+2) + 5324*a(n+3) - 902*a(n+4)) + a(n+3)*(-374*a(n+3) + 83*a(n+4))) + a(n+2)*(a(n+2)*(-5*a(n+2) - 176*a(n+3) + 32*a(n+4)) + a(n+3)*(+17*a(n+3) - 4*a(n+4))) for all n in Z. - _Michael Somos_, Aug 06 2016
%F A005258 a(n) = binomial(2*n, n)*hypergeom([-n, -n, -n],[1, -2*n], 1). - _Peter Luschny_, Feb 10 2018
%F A005258 a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*binomial(n+k,k)^2. - _Peter Bala_, Feb 10 2018
%F A005258 G.f. y=A(x) satisfies: 0 = x*(x^2 + 11*x - 1)*y'' + (3*x^2 + 22*x - 1)*y' + (x + 3)*y. - _Gheorghe Coserea_, Jul 01 2018
%F A005258 From _Peter Bala_, Jan 15 2020: (Start)
%F A005258 a(n) = Sum_{0 <= j, k <= n} (-1)^(j+k)*C(n,k)*C(n+k,k)^2*C(n,j)* C(n+k+j,k+j).
%F A005258 a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)^2*C(n+k,k)*C(n,j)* C(n+k+j,k+j).
%F A005258 a(n) = Sum_{0 <= j, k <= n} (-1)^j*C(n,k)^2*C(n,j)*C(3*n-j-k,2*n). (End)
%F A005258 a(n) = [x^n] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^m at m = 1. At m = 2 we get the Apéry numbers A005259. - _Peter Bala_, Dec 22 2020
%F A005258 a(n) = (-1)^n*Sum_{j=0..n} (1 - 5*j*H(j) + 5*j*H(n - j))*binomial(n, j)^5, where H(n) denotes the n-th harmonic number, A001008/A002805. (Paule/Schneider). - _Peter Luschny_, Jul 23 2021
%F A005258 From _Bradley Klee_, Jun 05 2023: (Start)
%F A005258 The g.f. T(x) obeys a period-annihilating ODE:
%F A005258 0=(3 + x)*T(x) + (-1 + 22*x + 3*x^2)*T'(x) + x*(-1 + 11*x + x^2)*T''(x).
%F A005258 The periods ODE can be derived from the following Weierstrass data:
%F A005258 g2 = 3*(1 - 12*x + 14*x^2 + 12*x^3 + x^4);
%F A005258 g3 = 1 - 18*x + 75*x^2 + 75*x^4 + 18*x^5 + x^6;
%F A005258 which determine an elliptic surface with four singular fibers. (End)
%F A005258 Conjecture: a(n)^2 = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A143007(n, k). - _Peter Bala_, Jul 08 2024
%e A005258 G.f. = 1 + 3*x + 19*x^2 + 147*x^3 + 1251*x^4 + 11253*x^5 + 104959*x^6 + ...
%p A005258 with(combinat): seq(add((multinomial(n+k,n-k,k,k))*binomial(n,k), k=0..n), n=0..18); # _Zerinvary Lajos_, Oct 18 2006
%p A005258 a := n -> binomial(2*n, n)*hypergeom([-n, -n, -n], [1, -2*n], 1):
%p A005258 seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Feb 10 2018
%t A005258 a[n_] := HypergeometricPFQ[ {n+1, -n, -n}, {1, 1}, 1]; Table[ a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jan 20 2012, after _Vladeta Jovovic_ *)
%t A005258 Table[Sum[Binomial[n,k]^2 Binomial[n+k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Aug 25 2019 *)
%o A005258 (Haskell)
%o A005258 a005258 n = sum [a007318 n k ^ 2 * a007318 (n + k) k | k <- [0..n]]
%o A005258 -- _Reinhard Zumkeller_, Jan 04 2013
%o A005258 (PARI) {a(n) = if( n<0, -(-1)^n * a(-1-n), sum(k=0, n, binomial(n, k)^2 * binomial(n+k, k)))} /* _Michael Somos_, Sep 18 2013 */
%o A005258 (GAP) a:=n->Sum([0..n],k->(-1)^(n-k)*Binomial(n,k)*Binomial(n+k,k)^2);;
%o A005258 A005258:=List([0..20],n->a(n));; # _Muniru A Asiru_, Feb 11 2018
%o A005258 (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k))); # _Muniru A Asiru_, Jul 29 2018
%o A005258 (Magma) [&+[Binomial(n,k)^2 * Binomial(n+k,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Nov 28 2018
%o A005258 (Python)
%o A005258 def A005258(n):
%o A005258     m, g = 1, 0
%o A005258     for k in range(n+1):
%o A005258         g += m
%o A005258         m *= (n+k+1)*(n-k)**2
%o A005258         m //= (k+1)**3
%o A005258     return g # _Chai Wah Wu_, Oct 02 2022
%Y A005258 Cf. A002736, A005259, A005429, A005430, A108625, A143413, A218690, A218692.
%Y A005258 Cf. A007318.
%Y A005258 Cf. A001008, A002805.
%Y A005258 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 A005258 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.
%K A005258 nonn,easy,nice
%O A005258 0,2
%A A005258 _N. J. A. Sloane_