A005258 - OEIS

%I M3057 #276 Jan 05 2025 19:51:33

%S 1,3,19,147,1251,11253,104959,1004307,9793891,96918753,970336269,

%T 9807518757,99912156111,1024622952993,10567623342519,109527728400147,

%U 1140076177397091,11911997404064793,124879633548031009,1313106114867738897,13844511065506477501

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

%C This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004

%C Equals the main diagonal of square array A108625. - _Paul D. Hanna_, Jun 14 2005

%C This sequence is t_5 in Cooper's paper. - _Jason Kimberley_, Nov 25 2012

%C 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 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%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

%D S. Melczer, An Invitation to Analytic Combinatorics, 2021; p. 129.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Simon Plouffe, <a href="/A005258/b005258.txt">Table of n, a(n) for n = 0..954</a>

%H 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 R. 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 R. 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 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 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 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 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 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 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 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 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>

%H 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 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 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 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 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 R. K. Guy, <a href="/A005258/a005258.pdf">Letter to N. J. A. Sloane, Oct 1985</a>

%H 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 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 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 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.

%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 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, 2016, 2:5.

%H 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 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 Simon Plouffe, <a href="http://plouffe.fr/OEIS/b005258.txt">The first 2553 Apéry numbers</a>

%H 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AperyNumber.html">Apéry Number.</a>

%H 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 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 a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - _Vladeta Jovovic_, Apr 24 2003

%F 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 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 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 a(n) ~ ((11+5*sqrt(5))/2)^(n+1/2)/(2*Pi*5^(1/4)*n). - _Vaclav Kotesovec_, Oct 05 2012

%F 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 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 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 a(n) = binomial(2*n, n)*hypergeom([-n, -n, -n],[1, -2*n], 1). - _Peter Luschny_, Feb 10 2018

%F a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*binomial(n+k,k)^2. - _Peter Bala_, Feb 10 2018

%F 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 From _Peter Bala_, Jan 15 2020: (Start)

%F 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 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 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 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 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 From _Bradley Klee_, Jun 05 2023: (Start)

%F The g.f. T(x) obeys a period-annihilating ODE:

%F 0=(3 + x)*T(x) + (-1 + 22*x + 3*x^2)*T'(x) + x*(-1 + 11*x + x^2)*T''(x).

%F The periods ODE can be derived from the following Weierstrass data:

%F g2 = 3*(1 - 12*x + 14*x^2 + 12*x^3 + x^4);

%F g3 = 1 - 18*x + 75*x^2 + 75*x^4 + 18*x^5 + x^6;

%F which determine an elliptic surface with four singular fibers. (End)

%F 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 G.f. = 1 + 3*x + 19*x^2 + 147*x^3 + 1251*x^4 + 11253*x^5 + 104959*x^6 + ...

%p 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 a := n -> binomial(2*n, n)*hypergeom([-n, -n, -n], [1, -2*n], 1):

%p seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Feb 10 2018

%t 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 Table[Sum[Binomial[n,k]^2 Binomial[n+k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Aug 25 2019 *)

%o (Haskell)

%o a005258 n = sum [a007318 n k ^ 2 * a007318 (n + k) k | k <- [0..n]]

%o -- _Reinhard Zumkeller_, Jan 04 2013

%o (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 (GAP) a:=n->Sum([0..n],k->(-1)^(n-k)*Binomial(n,k)*Binomial(n+k,k)^2);;

%o A005258:=List([0..20],n->a(n));; # _Muniru A Asiru_, Feb 11 2018

%o (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k))); # _Muniru A Asiru_, Jul 29 2018

%o (Magma) [&+[Binomial(n,k)^2 * Binomial(n+k,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Nov 28 2018

%o (Python)

%o def A005258(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)**3

%o     return g # _Chai Wah Wu_, Oct 02 2022

%Y Cf. A002736, A005259, A005429, A005430, A108625, A143413, A218690, A218692.

%Y Cf. A007318.

%Y Cf. A001008, A002805.

%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.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_