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 A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2. (Formerly M4020) 86
 1, 5, 73, 1445, 33001, 819005, 21460825, 584307365, 16367912425, 468690849005, 13657436403073, 403676083788125, 12073365010564729, 364713572395983725, 11111571997143198073, 341034504521827105445, 10534522198396293262825, 327259338516161442321485 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Prime Apéry 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, May 22 2005 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 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 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 REFERENCES R. Apéry, "Interpolation de fractions continues et irrationalité de certaines constantes," in Mathématiques, Ministère universités (France), Comité travaux historiques et scientifiques. Bull. Section Sciences, Vol. 3, pp. 243-246, 1981. F. Beukers, Consequences of Apéry's work on zeta(3), 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] Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153. W. Koepf, Hypergeometric Summation, Vieweg, 1998, p. 146. 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. Lipshitz, Leonard, and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..656 (first 101 terms from T. D. Noe) B Adamczewski, JP Bell, E Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016. J.-P. Allouche, A remark on Apéry's numbers, J. Comput. Appl. Math. 83 (1997), 123-125. R. Apéry, Irrationalité de zeta(2) et zeta(3), 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. R. Apéry, Sur certaines séries entières arithmétiques, Groupe de travail d'analyse ultramétrique, 9 no. 1 (1981-1982), Exp. No. 16, 2 p. Thomas Baruchel, C Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016. F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210. Francis Brown, Irrationality proofs for zeta values, moduli spaces and dinner parties, arXiv:1412.6508 [math.NT], 2014. Y. C. Chen, Q.-H. Hou, Y-P. Mu, A telescoping method for double summations, J. Comp. Appl. Math. 196 (2006) 553-566, Example 4. M. Coster, Email, Nov 1990 E. Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013. E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004. E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215. G. A. Edgar, A formula with Legendre polynomials, Sci. Math. Research posting Mar 21 2005 C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. C. Elsner, On prime-detecting sequences from Apéry's recurrence formulas for zeta(3) and zeta(2), JIS 11 (2008) 08.5.1 S. Fischler, Irrationalité de valeurs de zeta, arXiv:math/0303066 [math.NT], 2003. S. Garrabrant, I. Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014. R. K. Guy, Letter to N. J. A. Sloane, Oct 1985 V. Kotesovec, Asymptotic of generalized Apéry sequences with powers of binomial coefficients, Nov 04 2012 L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic Ji-Cai Liu, Supercongruences for the (p-1)th Apéry number, arXiv:1803.11442 [math.NT], 2018. Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5 Stephen Melczer and Bruno Salvy, Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables, arXiv:1605.00402 [cs.SC], 2016. R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014. Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers Math Overflow, A conjectured formula for Apéry numbers E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013. A. Strangeway, A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space, arXiv preprint arXiv:1302.5089 [math.AG], 2013. A. Strangeway, Quantum reconstruction for Fano bundles on projective space, Nagoya Math. J., Volume 218 (2015), 1-28. V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp. Volker Strehl, Binomial identities -- combinatorial and algorithmic aspects, Discrete Mathematics, Vol. 136 (1994), 309-346. Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018. Z.-W. Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011. Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory 132(2012), 2673-2699. [Zhi-Wei Sun, Mar 21 2013] Z.-W. Sun, On sums of Apéry polynomials and related congruences, arXiv:1101.1946 [math.NT], 2011-2014. [Zhi-Wei Sun, Mar 21 2013] A. van der Poorten, A proof that Euler missed ..., Math. Intelligencer 1,4, December 1979, pp. 196-203, (b_n) after eq. (1.2), and Exercise 3. Chen Wang, Two congruences concerning Apéry numbers, arXiv:1909.08983 [math.NT], 2019. Eric Weisstein's World of Mathematics, Apéry Number Eric Weisstein's World of Mathematics, Strehl Identities Eric Weisstein's World of Mathematics, Schmidt's Problem E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3. FORMULA (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1. 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 a(n) = Sum_{k >= 0} A063007(n, k)*A000172(k)). A000172 = Franel numbers. - Philippe Deléham, Aug 14 2003 G.f.: (-1/2)*(3*x - 3 + (x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],,(-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 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 a(n) ~ (1+sqrt(2))^(4*n+2)/(2^(9/4)*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 01 2012 a(n) = sum_{k=0..n} C(n,k)^2 * C(n+k,k)^2. - Joerg Arndt, May 11 2013 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 EXAMPLE G.f. = 1 + 5*x + 73*x^2 + 1445*x^3 + 33001*x^4 + 819005*x^5 + 21460825*x^6 + ... 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 MAPLE 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; MATHEMATICA Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1, 1, 1}, 1], {n, 0, 13}] (* Jean-François Alcover, Apr 01 2011 *) Table[Sum[(Binomial[n, k]Binomial[n+k, k])^2, {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Oct 15 2011 *) 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 *) PROG (PARI) a(n)=sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2) \\ Charles R Greathouse IV, Nov 20 2012 (Haskell) a005259 n = a005259_list !! n a005259_list = 1 : 5 : zipWith div (zipWith (-)    (tail \$ zipWith (*) a006221_list a005259_list)    (zipWith (*) (tail a000578_list) a005259_list)) (drop 2 a000578_list) -- Reinhard Zumkeller, Mar 13 2014 (GAP) List([0..20], n->Sum([0..n], k->Binomial(n, k)^2*Binomial(n+k, k)^2)); # Muniru A Asiru, Sep 28 2018 CROSSREFS Cf. A002736, A005258, A005259, A005429, A005430, A059415, A059416, A063007, A000172. Cf. A006221, A000578, A006353. Related to diagonal of rational functions: A268545-A268555. 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.) 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. Sequence in context: A222352 A159509 A127167 * A195636 A213111 A062440 Adjacent sequences:  A005256 A005257 A005258 * A005260 A005261 A005262 KEYWORD nonn,easy,nice AUTHOR Simon Plouffe, N. J. A. Sloane, May 20 1991 STATUS approved

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Last modified October 23 20:14 EDT 2019. Contains 328373 sequences. (Running on oeis4.)