login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
(Formerly M4020)
36
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

REFERENCES

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. Asterisque, 61 (1979), 11-13.

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 retombees", Rencontres Arithmetiques de Caen, June 2-3, 1995 [Mentions divisibility of a(n) by powers of 5 and powers of 11]

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.

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

J.-P. Allouche, A remark on Apéry's numbers, J. Comput. Appl. Math. 83 (1997), 123-125.

R. Apéry, Sur certaines séries entières arithmétiques, Groupe de travail d'analyse ultramétrique, 9 no. 1 (1981-1982), Exp. No. 16, 2 p.

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.

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, 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, Irrationalite de valeurs de zeta, arXiv:math/0303066 [math.NT], 2003.

S. Garrabrant, I. Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.

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

Stephen Melczer and Bruno Salvy, Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables, arXiv:1605.00402, 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

Volker Strehl, Binomial identities -- combinatorial and algorithmic aspects, Discrete Mathematics, Vol. 136 (1994), 309-346.

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]

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],(-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 Edgar conjectures 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

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

EXAMPLE

G.f. = 1 + 5*x + 73*x^2 + 1445*x^3 + 33001*x^4 + 819005*x^5 + 21460825*x^6 + ...

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

CROSSREFS

Cf. A002736, A005258, A005259, A005429, A005430, A059415, A059416, A063007, A000172.

Cf. A006221, A000578.

Sequence in context: A222352 A159509 A127167 * A195636 A213111 A062440

Adjacent sequences:  A005256 A005257 A005258 * A005260 A005261 A005262

KEYWORD

nonn,easy,nice,changed

AUTHOR

Simon Plouffe, N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 1 01:17 EDT 2016. Contains 274319 sequences.