A059415 -id:A059415 - cp's OEIS frontend

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

1, 5, 73, 1445, 33001, 819005, 21460825, 584307365, 16367912425, 468690849005, 13657436403073, 403676083788125, 12073365010564729, 364713572395983725, 11111571997143198073, 341034504521827105445, 10534522198396293262825, 327259338516161442321485

Offset: 0

Simon Plouffe, N. J. A. Sloane, May 20 1991

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
Named after the French mathematician Roger Apéry (1916-1994). - Amiram Eldar, Jun 10 2021

			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

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.
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.
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.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..656 (first 101 terms from T. D. Noe)
Boris Adamczewski, Jason P. Bell and Eric Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Jean-Paul Allouche, A remark on Apéry's numbers, J. Comput. Appl. Math., Vol. 83 (1997), pp. 123-125.
Roger 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, Vol. 61 (1979), pp. 11-13.
Roger Apéry, Sur certaines séries entières arithmétiques, Groupe de travail d'analyse ultramétrique, Vol. 9, No. 1 (1981-1982), Exp. No. 16, 2 p.
Roger Apéry, Interpolation de fractions continues et irrationalité de certaines constantes, Bulletin de la section des sciences du C.T.H.S III (1981), pp. 37-53.
Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Frits Beukers, Another congruence for the Apéry numbers, J. Number Theory, Vol. 25, No. 2 (1987), pp. 201-210.
Frits 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]
Francis Brown, Irrationality proofs for zeta values, moduli spaces and dinner parties, arXiv:1412.6508 [math.NT], 2014.
William Y. C. Chen, Qing-Hu Hou and Yan-Ping Mu, A telescoping method for double summations, J. Comp. Appl. Math., Vol. 196, No. 2 (2006), pp. 553-566, Example 4.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
M. Coster, Email, Nov 1990
Eric Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
Gerald A. Edgar, A formula with Legendre polynomials, Sci. Math. Research posting Mar 21 2005.
G. A. Edgar, The Apéry Numbers as a Stieltjes Moment Sequence, arXiv:2005.10733 [math.CA], 2020.
Carsten Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., Vol. 43, No. 1 (2005), pp. 31-45.
Carsten Elsner, On prime-detecting sequences from Apéry's recurrence formulas for zeta(3) and zeta(2), JIS, Vol. 11 (2008), Article 08.5.1.
Stéphane Fischler, Irrationalité de valeurs de zeta, arXiv:math/0303066 [math.NT], 2003.
Scott Garrabrant and Igor Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
Ira Gessel, Some congruences for Apéry numbers, Journal of Number Theory, Vol. 14, No. 3 (1982) 362-368.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See gamma p. 3.
Richard K. Guy, Letter to N. J. A. Sloane, Oct 1985
Vaclav Kotesovec, Asymptotic of generalized Apéry sequences with powers of binomial coefficients, Nov 04 2012
Antoine Labelle et al., A formula for Apéry numbers, MathOverflow, 2025.
Leonard Lipshitz and Alfred J. van der Poorten, Rational functions, diagonals, automata and arithmetic, 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; alternative link; Wayback Machine copy.
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, Vol. 2 (2016), Article 5.
Stephen Melczer and Bruno Salvy, Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables, arXiv:1605.00402 [cs.SC], 2016.
Romeo Meštrović, 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, Funct. Approx. Comment. Math., Vol. 48, No. 1 (2013), pp. 29-36; alternative link.
Math Overflow, A conjectured formula for Apéry numbers
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.
Eric Rowland, Reem Yassawi and Christian Krattenthaler, Lucas congruences for the Apéry numbers modulo p^2, arXiv:2005.04801 [math.NT], 2020.
Andrew Strangeway, A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space, arXiv preprint arXiv:1302.5089 [math.AG], 2013.
Andrew Strangeway, Quantum reconstruction for Fano bundles on projective space, Nagoya Math. J., Vol. 218 (2015), pp. 1-28.
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.
Volker 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.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Zhi-Wei Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011.
Zhi-Wei Sun, On sums of Apéry polynomials and related congruences, J. Number Theory 132(2012), 2673-2699. [_Zhi-Wei Sun_, Mar 21 2013]
Zhi-Wei Sun. Sun, On sums of Apéry polynomials and related congruences, arXiv:1101.1946 [math.NT], 2011-2014. [_Zhi-Wei Sun_, Mar 21 2013]
Alfred van der Poorten, A proof that Euler missed ..., Math. Intelligencer, Vol. 1, No. 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.
Ernest X. W. Xia and Olivia X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, Vol. 20 (2013), #P3.

Apéry's number or Apéry's constant zeta(3) is A002117. - N. J. A. Sloane, Jul 11 2023
Cf. A002736, A005258, 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.
Cf. A092826 (prime terms).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k)^2)); # Muniru A Asiru, Sep 28 2018

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

[&+[Binomial(n, k) ^2 *Binomial(n+k, k)^2: k in [0..n]]:n in  [0..17]]; // Marius A. Burtea, Jan 20 2020

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;
# Alternative:
a := n -> hypergeom([-n, -n, 1+n, 1+n], [1, 1, 1], 1):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Jan 19 2020

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 *)

a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k,k))^2) \\ Charles R Greathouse IV, Nov 20 2012

def A005259(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m *= ((n+k+1)*(n-k))**2
        m //=(k+1)**4
    return g # Chai Wah Wu, Oct 02 2022

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.
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 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
From Peter Bala, Jan 18 2020: (Start)
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).
a(n) = Sum_{0 <= j, k <= n} C(n,k) * C(n+k,k) * C(k,j)^3 (see Koepf, p. 55).
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).
Diagonal coefficients of the rational function 1/((1 - x - y)*(1 - z - t) - x*y*z*t) (Straub, 2014). (End)
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
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n, k). - Peter Bala, Jul 18 2024
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

A156164 Decimal expansion of 17 + 12*sqrt(2).

Original entry on oeis.org

3, 3, 9, 7, 0, 5, 6, 2, 7, 4, 8, 4, 7, 7, 1, 4, 0, 5, 8, 5, 6, 2, 0, 2, 6, 4, 6, 9, 0, 5, 1, 6, 3, 7, 6, 9, 4, 2, 8, 3, 6, 0, 6, 2, 5, 0, 4, 5, 2, 3, 3, 7, 6, 8, 7, 8, 1, 2, 0, 1, 5, 6, 8, 5, 5, 8, 8, 8, 7, 8, 9, 7, 4, 1, 5, 4, 5, 2, 8, 4, 4, 6, 6, 2, 0, 4, 6, 5, 0, 4, 1, 1, 9, 3, 1, 6, 9, 8, 8, 7, 2, 8, 2, 0, 1

Offset: 2

Klaus Brockhaus, Feb 09 2009

Lim_{n -> infinity} b(n)/b(n-1) = 17+12*sqrt(2) for b = A156160, A156161, A156162, A278310.

Conjecturally, the fractional part 0.97056 27484 ... of this constant equals ( (1 + 2 * Sum_{n >= 1} (-1)^n*exp(-2*Pi*n^2))/(1 + 2 * Sum_{n >= 1} exp(-2*Pi*n^2)) )^4. The series are rapidly converging. For example, summing both series from n = 1 to n = 2 approximates the fractional part of the constant as ( (1 - 2*exp(-2*Pi) + 2*exp(-8*Pi))/(1 + 2*exp(-2*Pi) + 2*exp(-8*Pi)) )^4 = 0.97056 27484 77140 58562 026(89) ..., correct to 23 decimal places. - Peter Bala, Jun 05 2019

			17 + 12*sqrt(2) = 33.97056274847714058562026469051637694283606250452337687...

G. C. Greubel, Table of n, a(n) for n = 2..10000
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.4. Example 4.
Index entries for algebraic numbers, degree 2

Cf. A002193: decimal expansion of sqrt(2); A156035: decimal expansion of 3+2*sqrt(2); A156163: decimal expansion of (19+6*sqrt(2))/17.

Cf. A005259, A059415/A059416.

RealDigits[17 + 12 Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, Nov 30 2014 *)

17+sqrt(288) \\ Charles R Greathouse IV, May 07 2015

17+12*sqrt(2) = (3+2*sqrt(2))^2 = (1+sqrt(2))^4. - Klaus Brockhaus, Feb 14 2009. (corrected by Bruno Berselli, Feb 19 2013)

A059416 Denominators of sequence arising from Apery's proof that zeta(3) is irrational.

Original entry on oeis.org

1, 1, 4, 36, 288, 36000, 800, 1372000, 2195200, 2667168000, 2667168000, 28400004864, 3550000608000, 311974053431040, 7799351335776000, 7799351335776000, 1134451103385600, 306545704901339904000, 6812126775585331200, 233621887768698933504000

Offset: 0

N. J. A. Sloane, Jan 30 2001

			0, 6, 351/4, 62531/36, ...

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.

Seiichi Manyama, Table of n, a(n) for n = 0..768
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

Cf. A059415, A005259.

a := proc(n) option remember; if n=0 then 0 elif n=1 then 6 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;

a[n_] := Sum[ Binomial[n, k]^2*Binomial[k + n, k]^2*(Sum[1/m^3, {m, 1, n}] + Sum[(-1)^(m - 1)/(2*m^3*Binomial[n, m]*Binomial[m + n, m]), {m, 1, k}]), {k, 0, n}]; Table[a[n] // Denominator, {n, 0, 19}] (* Jean-François Alcover, Jul 16 2013, from the non-recursive 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.

A257045 Denominators of Apéry's rational approximations p_n/q_n to zeta(3).

Original entry on oeis.org

1, 5, 292, 52020, 9504288, 29484180000, 17168660000, 801669704780000, 35930841355360000, 1250077234358967840000, 36426677336311407264000, 11464402743063221545440000, 42860453128110714373355232000

Offset: 0

Jean-François Alcover, Apr 15 2015

			0, 6/5,  351/292,  62531/52020, 11424695/9504288, 35441662103/29484180000, ...
0, 1.2, 1.202..., 1.2020569...,   1.202056903...,        1.20205690316..., ...

Seiichi Manyama, Table of n, a(n) for n = 0..358
Stéphane Fischler, Irrationalité de valeurs de zêta. Séminaire Bourbaki (2002-2003) Vol. 45, pp. 27-62 [in French]
Eric Weisstein's MathWorld, Apéry's Constant
Wikipedia, Apéry's constant

Cf. A002117, A005259, A059415, A059416.

p[n_] := Sum[Binomial[n+k, k]^2*Binomial[n, k]^2*(Sum[1/m^3, {m, 1, n}] + Sum[ (-1)^(m-1)/(2*m^3*Binomial[n, m]*Binomial[m+n, m]), {m, 1, k}]), {k, 0, n}]; q[n_] := Sum[Binomial[n+k, k]^2*Binomial[n, k]^2, {k, 0, n}]; Table[p[n]/q[n], {n, 0, 12}] // Denominator

See Mathematica script.

A288485 Expansion of (E_4(q) - 28E_4(q^2) + 63E_4(q^3) - 36*E(q^6)) / 240.

Original entry on oeis.org

1, -19, 91, -179, 126, -1, 344, -1459, 2521, -2394, 1332, -737, 2198, -6536, 11466, -11699, 4914, 485, 6860, -22554, 31304, -25308, 12168, -6625, 15751, -41762, 68131, -61576, 24390, -126, 29792, -93619, 121212, -93366, 43344, -15803, 50654, -130340

Offset: 1

Seiichi Manyama, Jun 09 2017

sign

Define f(q) = (eta(q^2)*eta(q^3))^7/(eta(q)*eta(q^6))^5, g(q) = Sum_{n>=1} a(n)/n^3 * q^n and t(q) = (eta(q)*eta(q^6)/(eta(q^2)*eta(q^3)))^12.
And define the sequence {b(n)} = {0, 6, 351/4, 62531/36, ...} as the solutions of the recursion (n+1)^3*b(n+1) = (34*n^3 + 51*n^2 + 27*n + 5)*b(n) - n^3*b(n-1), n >= 1 with b(0) = 0, b(1) = 6.
The following equation holds: 6*f(q)*g(q) = Sum_{n>=0} b(n)*t(q)^n.

			6*f(q)*g(q)
= 6*(1 + 5*q + 13*q^2 + 23*q^3 + 29*q^4 + 30*q^5 + 31*q^6 + 40*q^7 + ... )
  *(q - 19/8*q^2 + 91/27*q^3 - 179/64*q^4 + 126/125*q^5 - 1/216*q^6 + 344/343q^7 - ... )
= 6*q + 63/4*q^2 + 971/36*q^3 + 10679/288*q^4 + 1126103/36000*q^5 + 105401/2400*q^6 + 536870027/12348000*q^7 + ...
= 6 * (q - 12*q^2 + 66*q^3 - 220*q^4 + 495*q^5 - 804*q^6 + 1068*q^7 - ... )
  + 351/4 * (q^2 - 24*q^3 + 276*q^4 - 2024*q^5 + 10626*q^6 - 42528*q^7 + ... )
  + 62531/36 * (q^3 - 36*q^4 + 630*q^5 - 7140*q^6 + 58905*q^7 - ... )
  + 11424695/288 * (q^4 - 48*q^5 + 1128*q^6 - 17296*q^7 + ... )
  + 35441662103/36000 * (q^5 - 60*q^6 + 1770*q^7 - ... )
  + ...

D. Zagier, "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Wikipedia, Apéry's theorem.

{b(n)} = {A059415(n)/A059416(n)} = {0, 6, 351/4, 62531/36, ...}.

Cf. A001158 (sigma_3(n)), A004009 (E_4), A006353 (f(q)), A226235 (t(q)).

a[n_Integer] := Module[{b, ary}, b = Join[{0}, Table[DivisorSigma[3, i], {i, 1, n}]]; ary = b; Do[If[Mod[i, 2] == 0, ary[[i + 1]] -= 28*b[[i/2 + 1]]]; If[Mod[i, 3] == 0, ary[[i + 1]] += 63*b[[i/3 + 1]]]; If[Mod[i, 6] == 0, ary[[i + 1]] -= 36*b[[i/6 + 1]]];, {i, 1, n}]; Rest[ary]]; a[38] (* Robert P. P. McKone, Aug 23 2023 *)

def A001158(n)
  s = 0
  (1..n).each{|i| s += i * i * i if n % i == 0}
  s
end
def A288485(n)
  a = [0] + (1..n).map{|i| A001158(i)}
  ary = a.clone
  (1..n).each{|i|
    ary[i] -= 28 * a[i / 2] if i % 2 == 0
    ary[i] += 63 * a[i / 3] if i % 3 == 0
    ary[i] -= 36 * a[i / 6] if i % 6 == 0
  }
  ary[1..-1]
end
p A288485(100)

A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

Original entry on oeis.org

0, 1, 5, 9, 29, 115, 251, 65, 5191, 1039, 2035, 10391, 2077, 72703, 58157, 256103, 259703, 1817471, 1817521, 7270009, 1454021, 28567, 67323, 25243, 389467, 21810107, 47982293, 6854599, 9822481, 9895981, 11132213, 66793523, 11755653433, 2351131157, 30564700141, 30564710941, 78708473, 237497419, 237487619, 23511313481, 23511309071, 61129406407, 5557218637, 61129406447, 244517610353

Offset: 0

Wolfdieter Lang, May 16 2018

The corresponding denominators are given in A303989.
The numerators of the rational triangle c_{n,k} are denoted by T(n,k). The triangle c_{n,k} is used to compute Apéry's sequence of rationals a_n = A059415(n)/A059416(n), satisfying a certain three term recurrence, as a(n) = Sum_{k=0..n} c_{n,k}*(binomial(n+k,k)*binomial(n,k))^2 = Sum_{k=0..n} (T(n,k)/A303989(n,k))*A303987(n,k).
The column k = 0 gives the numerators of Zeta3(n) = A007408(n)/A007409(n), with Zeta3(0) := 0.

			The triangle T(n, k) begins:
  n/k      0       1        2        3           4          5           6
  0:       0
  1:       1       5
  2:       9      29      115
  3:     251      65     5191     1039
  4:    2035   10391     2077    72703       58157
  5:  256103  259703  1817471  1817521     7270009    1454021
  6:   28567   67323    25243   389467    21810107   47982293     6854599
  ...
  row n = 7: 9822481 9895981 11132213 66793523 11755653433 2351131157 30564700141 30564710941,
  row n = 8: 78708473 237497419 237487619 23511313481 23511309071 61129406407 5557218637 61129406447 244517610353,
  row n = 9: 19148110939 19237016539 211601625329 211601801729 2750823224027 42320357851 550164649543 550164651163 37411196140169 37411196579209,
  ...
------------------------------------------------------------------------------
The rational triangle c_{n,k} starts:
  n\k        0            1              2                3               4
  0:        0/1
  1:        1/1          5/4
  2:        9/8         29/24         115/96
  3:      251/216       65/54        5191/4320        1039/864
  4:    2035/1728    10391/8640      2077/1728      72703/60480      58157/48384
  ...
  row n = 5:  256103/216000 259703/216000 1817471/1512000 1817521/1512000 7270009/6048000 1454021/1209600,
  ...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.

A. van der Poorten, A proof that Euler missed ..., Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/79), no. 4, 195-203, c_{n,k} in section 4.
Wikipedia, Apery's theorem

Cf. A005259, A007408/A007409, A059415, A059416, A063007, A303987, A303989.

T(n,k) = numerator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ Jason Yuen, May 27 2025

T(n,k) = numerator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n,m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n,m) = A063007(n,m).

cp's OEIS Frontend

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

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A059416 Denominators of sequence arising from Apery's proof that zeta(3) is irrational.

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A257045 Denominators of Apéry's rational approximations p_n/q_n to zeta(3).

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A288485 Expansion of (E_4(q) - 28*E_4(q^2) + 63*E_4(q^3) - 36*E(q^6)) / 240.

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A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

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A112637 a(n) = b(n)*lcm{1,2,..,n}^3 where b(n) is a sequence of rationals arising in irrationality proof of zeta(3).

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A288485 Expansion of (E_4(q) - 28E_4(q^2) + 63E_4(q^3) - 36*E(q^6)) / 240.