A002893 -id:A002893 - cp's OEIS frontend

A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.

1, 4, 40, 544, 8536, 145504, 2618176, 48943360, 941244376, 18502137184, 370091343040, 7508629231360, 154145664817600, 3196100636757760, 66834662101834240, 1407913577733228544, 29849617614785770456, 636440695668355742560, 13638210075999240396736, 293565508750164008207104, 6344596821114216520841536

Offset: 0

Hugo Pfoertner, Aug 06 2017

Sequence epsilon in Almkvist, Straten, Zudilin article.

Seiichi Manyama, Table of n, a(n) for n = 0..500
G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See epsilon p. 3.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Armin Straub and Wadim Zudilin, Positivity of rational functions and their diagonals, J. Approx. Theory, 195:57-69, 2015. arXiv:1312.3732 [math.NT].
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.
Jeremy Tan, Simplifying a binomial sum for bridge deals with specific voids, Mathematics Stack Exchange, 2024.

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.

Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 23 2017 *)
a[n_] := Binomial[2 n, n]^2 HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1/2 - n, 1/2 - n}, 1];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 10 2022 *)

C=binomial; a(n) = sum (k=0, n, C(n,k)^2 * C(k+k,n)^2);

a(-1)=0, a(0)=1, a(n+1) = ((2*n+1)*(12*n^2+12*n+4)*a(n)-16*n^3*a(n-1))/(n+1)^3.
a(n) = Sum_{k=ceiling(n/2)..n} binomial(n,k)^2*binomial(2*k,n)^2. [Gorodetsky] - Michel Marcus, Feb 25 2021
a(n) ~ 2^(2*n - 3/4) * (1 + sqrt(2))^(2*n+1) / (Pi*n)^(3/2). - Vaclav Kotesovec, Jul 10 2021
From Peter Bala, Apr 10 2022: (Start)
The g.f. is the diagonal of the rational function 1/(1 - (x + y + z + t) + 2*(x*y*z + x*y*t + x*z*t + y*z*t) + 4*x*y*z*t) (Straub and Zudilin)
The g.f. appears to be the diagonal of the rational function 1/(1 - x - y + z - t - 2*(x*z + y*z + z*t) + 4*(x*y*t + x*z*t) + 8*x*y*z*t).
If true, then a(n) = [(x*y*z)^n] ( (x + y + z + 1)*(x + y + z - 1)*(x + y - z - 1)*(x - y - z + 1) )^n . (End)
a(n) = binomial(2*n, n)^2 * hypergeom([1/2-n/2, 1/2-n/2, -n/2, -n/2], [1, 1/2-n, 1/2-n], 1). - Peter Luschny, Apr 10 2022
G.f.: hypergeom([1/8, 3/8],[1], 256*x^2 / (1 - 4*x)^4)^2 / (1 - 4*x). - Mark van Hoeij, Nov 12 2022
a(n) = [(w*x*y*z)^n] ((w+z)*(x+z)*(y+z)*(w+x+y+z))^n = Sum_{0 <= j <= i <= n} binomial(n,i)^2*binomial(i,j)^2*binomial(n+j,i). - Jeremy Tan, Mar 28 2024

A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2C(n,l)C(k,l)*C(k+l,n)).

Original entry on oeis.org

1, 3, 27, 309, 4059, 57753, 866349, 13492251, 216077787, 3536145057, 58875891777, 994150929951, 16984143140589, 293036113226223, 5098773125244483, 89368239352074309, 1576424378494272987, 27964450505226314673, 498550055166916502121

Offset: 0

Hugo Pfoertner, Aug 06 2017

Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.

Robert Israel, Table of n, a(n) for n = 0..779
G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See zeta p. 3.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 07 2017

Table[Sum[Sum[(Binomial[n, k]^2*Binomial[n, j] Binomial[k, j] Binomial[k + j, n]), {j, 0, n} ], {k, 0, n}], {n, 0, 18}] (* Michael De Vlieger, Aug 07 2017 *)

C=binomial;
a(n) = sum(k=0,n, sum(l=0,n, C(n,k)^2 * C(n,l) * C(k,l) * C(k+l,n) ));

a(0) = 1, a(1) = 3,
a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3.
a(n) ~ 3^(3*n/2 + 1) * (1+sqrt(3))^(2*n+1) / (2^(n + 5/2) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jul 10 2021
G.f.: hypergeom([1/12,5/12],[1],(12*x/(1-6*x-27*x^2))^3)^2/(1-6*x-27*x^2)^(1/2). - Mark van Hoeij, Nov 11 2022

A143003 a(0) = 0, a(1) = 1, a(n+1) = (2n+1)(n^2+n+5)a(n) - n^6a(n-1).

Original entry on oeis.org

0, 1, 21, 1091, 114520, 21298264, 6410456640, 2923097201856, 1920450126458880, 1747596822651334656, 2133806329230225408000, 3405545462439659704320000, 6950705677729940374290432000, 17807686090745585163974737920000

Offset: 0

Peter Bala, Jul 19 2008

This is the case m = 1 of the general recurrence a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+2*m^2+2*m+1)*a(n) - n^6*a(n-1) (we suppress the dependence of a(n) on m), which arises when accelerating the convergence of the series Sum_{k>=1} 1/k^3 for Apery's constant zeta(3). For other cases see A066989 (m=0), A143004 (m=2), A143005 (m=3) and A143006 (m=4).
The solution to the general recurrence may be expressed as a sum: a(n) = n!^3*p_m(n)*Sum_{k = 1..n} 1/(k^3*p_m(k-1)*p_m(k)), where p_m(x) = Sum_{k = 0..n} C(2*k,k)^2*C(n+k,2*k)*C(x+k,2*k) is a polynomial in x of degree 2*m.
The first few are p_0(x) = 1, p_1(x) = 2*x^2 + 2*x + 1, p_2(x) = (3*x^4 + 6*x^3 + 9*x^2 + 6*x + 2)/2 and p_3(x) = (10*x^6 + 30*x^5 + 85*x^4 + 120*x^3 + 121*x^2 + 66*x + 18)/18. For fixed n, the sequence [p_n(k)]k>=0 is the crystal ball sequence for the product lattice A_n x A_n. See A143007 for the table of values [p_n(k)] n,k >= 0. Observe that [p_n(n)] n >= 0 is the sequence of Apery numbers A005259.
The reciprocity law p_m(n) = p_n(m) holds for nonnegative integers m and n. In particular we have p_m(1) = 2*m^2 + 2*m + 1 and p_m(2) = (3*m^4 + 6*m^3 + 9*m^2 + 6*m + 2)/2.
The polynomial p_m(x) is the unique polynomial solution of the difference equation (x+1)^3*f(x+1) + x^3*f(x-1) = (2*x+1)*(x^2+x+2*m^2+2*m+1)*f(x), normalized so that f(0) = 1. The reciprocity law now yields the Apery-like recursion m^3*p_m(x) + (m-1)^3*p_(m-2)(x) = (2*m-1)*(m^2-m+1+2*x^2+2*x)*p_(m-1)(x).
The polynomial functions p_m(x) have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_m(x-1), m = 1,2,3,..., satisfy a Riemann hypothesis (adapt the proof of the lemma on p. 4 of [BUMP et al.]).
The general recurrence in the first paragraph above has a second solution b(n) = n!^3*p_m(n) with initial conditions b(0) = 1, b(1) = 2*m^2+2*m+1. Hence the behavior of a(n) for large n is given by lim_{n -> infinity} a(n)/b(n) = Sum_{k>=1} 1/(k^3*p_m(k-1)*p_m(k)) = 1/((2*m^2+2*m+1) - 1^6/(3*(2*m^2+2*m+3) - 2^6/(5*(2*m^2+2*m+7) - 3^6/(7*(2*m^2+2*m+13) - ...)))) = Sum_{k>=1} 1/(m+k)^3. The final equality follows from a result of Ramanujan; see [Berndt, Chapter 12, Entry 32(iii)].
For the corresponding results for the constant zeta(2) see A142995. For corresponding results for the constant log(2) see A142979 and A142992.

Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag.

Seiichi Manyama, Table of n, a(n) for n = 0..181
D. Bump, K. Choi, P. Kurlberg and J. Vaaler, A local Riemann hypothesis, I, Math. Zeit. 233, (2000), 1-19.

Cf. A066989, A008288, A108625, A142979, A142992, A142995, A143004, A143005, A143006, A143007.

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.)

p := n -> 2*n^2+2*n+1: a := n -> n!^3*p(n)*sum (1/(k^3*p(k-1)*p(k)), k = 1..n): seq(a(n), n = 0..14)

RecurrenceTable[{a[0]==0,a[1]==1,a[n+1]==(2n+1)(n^2+n+5)a[n]- n^6 a[n-1]}, a[n],{n,15}] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = n!^3*p(n)*Sum_{k = 1..n} 1/(k^3*p(k-1)*p(k)), where p(n) = 2*n^2 + 2*n + 1 = A001844(n).
Recurrence: a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+5)*a(n) - n^6*a(n-1).
The sequence b(n):= n!^3*p(n) satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 5. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 - ... - (n-1)^6/((2*n-1)*(n^2-n+5)))))), for n >= 2. The behavior of a(n) for large n is given by lim_{n -> infinity} a(n)/b(n) = Sum_{k>=1} 1/(k^3*(4*k^4 + 1)) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 - ... - n^6/((2*n+1)*(n^2+n+5) - ...))))) = zeta(3) - 1, where the final equality follows from a result of Ramanujan; see [Berndt, Chapter 12, Entry 32(iii) at x = 1].

A183204 Central terms of triangle A181544.

Original entry on oeis.org

1, 4, 48, 760, 13840, 273504, 5703096, 123519792, 2751843600, 62659854400, 1451780950048, 34116354472512, 811208174862904, 19481055861877120, 471822589361293680, 11511531876280913760, 282665135367572129040

Offset: 0

Paul D. Hanna, Dec 30 2010

The g.f. for row n of triangle A181544 is (1-x)^(3n+1)*Sum_{k>=0}C(n+k-1,k)^3*x^k.
This sequence is s_7 in Cooper's paper. - Jason Kimberley, Nov 06 2012
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y*z + x*y + x*z + y + z)). - Gheorghe Coserea, Jul 14 2016
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017

			Triangle A181544 begins:
(1);
1, (4), 1;
1, 20, (48), 20, 1;
1, 54, 405, (760), 405, 54, 1;
1, 112, 1828, 8464, (13840), 8464, 1828, 112, 1; ...

Seiichi Manyama, Table of n, a(n) for n = 0..702 (terms 0..499 from Jason Kimberley)
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J., December 2012, Volume 29, Issue 1, pp 163-183.
Shaun Cooper, Jesús Guillera, Armin Straub, and Wadim Zudilin, Crouching AGM, Hidden Modularity, arXiv:1604.01106 [math.NT], 5-April-2016.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s7 p. 3.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Robert Osburn, Armin Straub, and Wadim Zudilin, A modular supercongruence for 6F5: an Apéry-like story, arXiv:1701.04098 [math.NT], 2017.
Wadim Zudilin, A generating function of the squares of Legendre polynomials, preprint arXiv:1210.2493 [math.CA], 2012.

Cf. A108625, A181544, A183205, A245086.
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.)

P:=PolynomialRing(Integers()); C:=Binomial;
A183204:=func; // or directly:
A183204:=func;
[A183204(n):n in[0..16]]; // Jason Kimberley, Oct 29 2012

Table[Sum[Binomial[n,j]^2 * Binomial[2*j,n] * Binomial[j+n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 05 2015 *)

{a(n)=polcoeff((1-x)^(3*n+1)*sum(j=0, 2*n, binomial(n+j, j)^3*x^j), n)}

a(n) = [x^n] (1-x)^(3n+1) * Sum_{k>=0} C(n+k-1,k)^3*x^k.
a(n) = Sum_{j = 0..n} C(n,j)^2 * C(2*j,n) * C(j+n,j). [Formula of Wadim Zudilin provided by Jason Kimberley, Nov 06 2012]
1/Pi = sqrt(7) Sum_{n>=0} (-1)^n a(n) (11895n + 1286)/22^(3n+3). [Cooper, equation (41)] - Jason Kimberley, Nov 06 2012
G.f.: sqrt((1-13*x+(1-26*x-27*x^2)^(1/2))/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2)))*hypergeom([1/12,5/12],[1],13824*x^7/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2))^3)^2. - Mark van Hoeij, May 07 2013
a(n) ~ 3^(3*n+3/2) / (4 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Apr 05 2015
G.f. A(x) satisfies 1/(1+4*x)^2 * A( x/(1+4*x)^3 ) = 1/(1+2*x)^2 * A( x^2/(1+2*x)^3 ) [see Cooper, Guillera, Straub, Zudilin]. - Joerg Arndt, Apr 08 2016
a(n) = (-1)^n*binomial(3n+1,n)* 4F3({-n,n+1,n+1,n+1};{1,1,2(n+1)}; 1). - M. Lawrence Glasser, May 15 2016
Conjecture D-finite with recurrence: n^3*a(n) - (2*n-1)*(13*n^2-13*n+4)*a(n-1) - 3*(n-1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, May 15 2016
0 = (-x^2+26*x^3+27*x^4)*y''' + (-3*x+117*x^2+162*x^3)*y'' + (-1+86*x+186*x^2)*y' + (4+24*x)*y, where y is g.f. - Gheorghe Coserea, Jul 14 2016
From Jeremy Tan, Mar 14 2024: (Start)
The conjectured D-finite recurrence can be proved by Zeilberger's algorithm.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,n) * binomial(2*n-k,n) = [(w*x*y*z)^n] ((w+y)*(x+z)*(y+z)*(w+x+y+z))^n. (End)
a(n) = Sum_{0 <= j, k <= n} binomial(n, k)^2 * binomial(n, j)^2 * binomial(k+j, n) = Sum_{k = 0..n} binomial(n, k)^2 * A108625(n, k). - Peter Bala, Jul 08 2024
From Peter Bala, Sep 18 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n+k, k)^3*binomial(3*n+1, n-k). Cf A245086.
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*A143007(n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). (End)

A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.

Original entry on oeis.org

1, 8, 88, 1088, 14296, 195008, 2728384, 38879744, 561787864, 8206324928, 120929313088, 1794924383744, 26802975999424, 402298219288064, 6064992788397568, 91786654611673088, 1393772628452578264, 21227503080738294464, 324160111169327247424

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 8*x + 88*x^2 +  1088*x^3 + 14296*x^5 + 195008*x^5 + ... - _Michael Somos_, May 29 2023

M. Petkovsek et al., "A=B", Peters, p. ix of second printing.

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Vn.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Vn.
Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020.

Cf. A002894, A036915, A057703.
Cf. A007318, A036916, A036829.
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.)

a036917 n = sum $ map
   (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n]
-- Reinhard Zumkeller, May 24 2012

a[n_] := (16 (n - 1/2)(2*n^2 - 2*n + 1)a[n - 1] - 256(n - 1)^3 a[n - 2])/n^3; a[0] = 1; a[1] = 8; Array[a, 19, 0] (* Or *)
f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *)
lmt = 20; Take[ 4^Range[0, 2 lmt]*CoefficientList[ Series[(4/Pi^2) EllipticK[4 x^(1/2)]^2, {x, 0, lmt}], x^(1/2)], lmt] (* Robert G. Wilson v *)
a[n_] := HypergeometricPFQ[{1/2, 1/2, -n, -n}, {1, 1/2-n, 1/2-n}, 1] * 4^n * (2n-1)!!^2 / n!^2 (* Vladimir Reshetnikov, Mar 08 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* Michael Somos, May 30 2023 *)

for(n=0,25, print1(sum(k=0,n, (binomial(2*n-2*k,n-k) *binomial(2*k,k))^2), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)); /* Michael Somos, May 29 2023 */

a(n) = (16*(n-1/2)*(2*n^2-2*n+1)*a(n-1)-256*(n-1)^3*a(n-2))/n^3.
a(n) = Sum_{k=0..n} (C(2 * (n-k), n-k) * C(2 * k, k))^2. [corrected by Tito Piezas III, Oct 19 2010]
a(n) = hypergeom([1/2, 1/2, -n, -n], [1, 1/2-n, 1/2-n], 1) * 4^n * (2n-1)!!^2 / n!^2. - Vladimir Reshetnikov, Mar 08 2014
a(n) ~ 2^(4*n+1) * log(n) / (n*Pi^2) * (1 + (4*log(2) + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 28 2015
G.f. y=A(x) satisfies: 0 = x^2*(16*x - 1)^2*y''' + 3*x*(16*x - 1)*(32*x - 1)*y'' + (1792*x^2 - 112*x + 1)*y' + 8*(32*x - 1)*y. - Gheorghe Coserea, Jul 03 2018
G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - Vaclav Kotesovec, Oct 01 2019
It appears that a(n) is equal to the coefficient of (x*y*z*t)^n in the expansion of (1+x+y+z-t)^n * (1+x+y-z+t)^n * (1+x-y+z+t)^n * (1-x+y+z+t)^n. Cf. A000172. - Peter Bala, Sep 21 2021
G.f. y = A(x) satisfies 0 = x*(1 - 16*x)*(2*y''*y - y'*y') + 2*(1 - 32*x)*y*y' - 16*y*y. - Michael Somos, May 29 2023
Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - Michael Somos, May 30 2023
From Paul D. Hanna, Mar 25 2024: (Start)
G.f. ( Sum_{n>=0} binomial(2*n,n)^2 * x^n )^2.
G.f. Sum_{n>=0} binomial(2*n,n)^3 * x^n * (1 - 16*x)^n. (End)

Replaced complicated definition via a formula with simple generating function provided by Vladeta Jovovic, Dec 01 2003. Thanks to Paul D. Hanna for suggesting this. - N. J. A. Sloane, Mar 25 2024

A005802 Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e., 1234-avoiding permutations); vexillary permutations (i.e., 2143-avoiding).

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718, 21937062144834, 162958355218089, 1221225517285209, 9225729232653663, 70209849031116183, 537935616492552297

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also the dimension of SL(3)-invariants in V^n tensor (V^*)^n, where V is the standard 3-dimensional representation of SL(3) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
Also the number of doubly-alternating permutations of length 2n with no four-term increasing subsequence (i.e., 1234-avoiding doubly-alternating permutations). The doubly-alternating permutations (counted by sequence A007999) are those permutations w such that both w and w^(-1) have descent set {2, 4, 6, ...}. - Joel B. Lewis, May 21 2009
Any permutation without an increasing subsequence of length 4 has a decreasing subsequence of length >= n/3, where n is the length of the sequence, by the Erdős-Szekeres theorem. - Charles R Greathouse IV, Sep 26 2012
Also the number of permutations of length n simultaneously avoiding patterns 1324 and 3416725 (or 1324 and 3612745). - Alexander Burstein, Jan 31 2014

Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(e), p. 453.

Alois P. Heinz, Table of n, a(n) for n = 0..1060
Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, arXiv:1312.4487 [math.CO], 2014-2015.
Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, European Journal of Combinatorics 43 (2015), 131-164.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
A. Burstein and J. Pantone, Two examples of unbalanced Wilf-equivalence, arXiv preprint arXiv:1402.3842 [math.CO], 2014.
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Andrew R. Conway and Anthony J. Guttmann, On 1324-avoiding permutations, Advances in Applied Mathematics, Volume 64, March 2015, p. 50-69.
Andrew R. Conway and Anthony J. Guttmann, Counting occurrences of patterns in permutations, arXiv:2306.12682 [math.CO], 2023. See p. 18.
Tom Denton, Algebraic and Affine Pattern Avoidance, arXiv preprint arXiv:1303.3767 [math.CO], 2013.
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, MAA Focus, August/September 2015, pp. 33-34. [Annotated scanned copy]
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Steven Finch, Pattern-Avoiding Permutations [Cached copy at the Wayback Machine]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
O. Guibert and E. Pergola, Enumeration of vexillary involutions which are equal to their mirror/complement, Discrete Math., Vol. 224 (2000), pp. 281-287.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns, Advances in Applied Mathematics, Vol. 17, 1996, pp. 381-407.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
Erik Ouchterlony, Pattern avoiding doubly alternating permutations
Permutation Pattern Avoidance Library (PermPAL), 0123
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Anders Bjorner and Richard P. Stanley, A combinatorial miscellany
R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.

A column of A047888. See also A224318, A223034, A223905.
Column k=3 of A214015.
A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

a:= n-> 2*add(binomial(2*k,k)*(binomial(n,k))^2*(3*k^2+2*k+1-n-2*k*n)/ (k+1)^2/(k+2)/(n-k+1),k=0..n);
A005802:=rsolve({a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41)*a(n + 1) - (9*n^2 + 18*n + 9)*a(n)},a(n),makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

a[n_] := 2Sum[Binomial[2k, k]Binomial[n, k]^2(3k^2+2k+1-n-2k*n)/((k+1)^2(k+2)(n-k+1)), {k, 0, n}]
(* Second program:*)
a[0] = a[1] = 1; a[n_] := a[n] = ((10*n^2+2*n-3)*a[n-1] + (-9*n^2+18*n-9)* a[n-2])/(n+2)^2; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 20 2017 *)
Table[HypergeometricPFQ[{1/2, -1 - n, -n}, {2, 2}, 4] / (n+1), {n, 0, 25}] (* Vaclav Kotesovec, Jun 07 2021 *)

a(n)=2*sum(k=0,n,binomial(2*k,k)*binomial(n,k)^2*(3*k^2+2*k+1-n-2*k*n)/(k+1)^2/(k+2)/(n-k+1)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2 * Sum_{k=0..n} binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2 + 2*k+1 - n - 2*k*n)/((k+1)^2 * (k+2) * (n-k+1)).
(4*n^2 - 2*n + 1)*(n + 2)^2*(n + 1)^2*a(n) = (44*n^3 - 14*n^2 - 11*n + 8)*n*(n + 1)^2*a(n - 1) - (76*n^4 + 42*n^3 - 49*n^2 - 24*n + 24)*(n - 1)^2*a(n - 2) + 9*(4*n^2 + 6*n + 3)*(n - 1)^2*(n - 2)^2*a(n - 3). - Vladeta Jovovic, Jul 16 2004
a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41) a(n + 1) - (9*n^2 + 18*n + 9) a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
a(n) = ((18*n+45)*A002893(n) - (7+2*n)*A002893(n+1)) / (6*(n+2)^2). - Mark van Hoeij, Jul 02 2010
G.f.: (1+5*x-(1-9*x)^(3/4)*(1-x)^(1/4)*hypergeom([-1/4, 3/4],[1],64*x/((x-1)*(1-9*x)^3)))/(6*x^2). - Mark van Hoeij, Oct 25 2011
a(n) ~ 3^(2*n+9/2)/(16*Pi*n^4). - Vaclav Kotesovec, Jul 29 2013
a(n) = Sum_{k=0..n} binomial(2k,k)*binomial(n+1,k+1)*binomial(n+2,k+1)/((n+1)^2*(n+2)). [Conway and Guttmann, Adv. Appl. Math. 64 (2015) 50]
For n > 0, (n+2)^2*a(n) - n^2*a(n-1) = 4*A086618(n). - Zhi-Wei Sun, Nov 16 2017
a(n) = hypergeom([1/2, -1 - n, -n], [2, 2], 4) / (n+1). - Vaclav Kotesovec, Jun 07 2021

Additional comments from Emeric Deutsch, Dec 06 2000
More terms from Naohiro Nomoto, Jun 18 2001
Edited by Dean Hickerson, Dec 10 2002
More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

A143583 Apéry-like numbers: a(n) = (1/C(2n,n))Sum_{k=0..n} C(2k,k)C(4k,2k)C(2n-2k,n-k)C(4n-4k,2n-2k).

Original entry on oeis.org

1, 12, 164, 2352, 34596, 516912, 7806224, 118803648, 1818757924, 27972399792, 431824158864, 6686855325888, 103814819552016, 1615296581684928, 25180747436810304, 393189646497706752, 6148451986328464164, 96269310864931432368, 1509065592479205772304

Offset: 0

Peter Bala, Aug 25 2008

These numbers bear some analogy to the Apéry numbers A005258. They appear in the evaluation of the spectral zeta function of the non-commutative harmonic oscillator zeta_Q(s) at s = 2 and satisfy a recurrence relation similar to the one satisfied by the Apéry numbers.

			G.f. = 1 + 12*x + 164*x^2 + 2352*x^3 + 34596*x^4 + 516912*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
K. Kimoto and M. Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, arXiv:math/0603700 [math.NT], 2006; Kyushu J. Math. Vol. 60, 2006, 383-404.
Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Gn.
Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Gn.
Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020.

Cf. A005258.

a := n -> 1/binomial(2*n, n)*add(binomial(2*k, k)*binomial(4*k, 2*k)*binomial(2*n-2*k, n-k)*binomial(4*n-4*k, 2*n-2*k), k = 0..n): seq(a(n), n = 0..25);
series( 2*EllipticK(4*x^(1/2))/(Pi*sqrt(1-16*x)), x=0, 20); # Mark van Hoeij, Apr 06 2013
A143583 := n -> 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1):
seq(simplify(A143583(n)), n = 0..18); # Peter Luschny, Nov 12 2022

Table[1/Binomial[2*n,n]*Sum[Binomial[2*k,k]*Binomial[4*k,2*k]*Binomial[2*n-2*k,n-k]*Binomial[4*n-4*k,2*n-2*k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2013 *)

a(n) = (1/C(2n,n))*sum {k = 0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).
Recurrence relation:
a(0) = 1, a(1) = 12, n^2*a(n) = 4*(8*n^2-8*n+3)*a(n-1) - 256*(n-1)^2*a(n-2).
Congruences:
For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N.
a(n) ~ 16^n/(Pi*sqrt(Pi*n)) * (log(n) + gamma + 6*log(2)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 11 2013
a(n) = sum {k = 0..n} 4^(n-k) C(2k,k)^2*C(2n-2k,n-k). - Tito Piezas III, Dec 12 2014
a(n) = hypergeom([1/2,1/2,n+1],[1,n+3/2],1)*2^(5*n+1)*n!/((2*n+1)!!*Pi) - G. A. Edgar, Dec 10 2016
a(n) = binomial(4*n,2*n)*hypergeom([1/4,3/4,-n,-n], [1,1/4-n,3/4-n], 1). - Peter Luschny, May 14 2020
From Peter Luschny, Nov 12 2022: (Start)
a(n) = 16^n*Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2*binomial(n, k).
a(n) = 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1). (End)

A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0

Offset: 0

Michael Somos, Sep 22 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077.

			G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A005928, A006077, A113062, A180318, A227454, A227696.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))}

Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.
Expansion of 2*b(q^4) - b(q) = b(q^2)^3 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...].
Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696.
G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.
a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).

A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 4, 2, 1, 3, 6, 4, 0, -1, 4, 6, 4, 2, 2, 2, 3, 4, 2, 0, 1, 6, 8, 2, 0, 3, 6, 0, -2, 0, 6, 6, 4, 4, 2, 4, 3, 4, 0, -2, 0, 6, 8, 2, 2, -1, 6, 4, 2, 1, 4, 6, 4, 2, 0, 6, 0, 0, 0, 0, 4, 6, 8, 2, 1, 2, 12, 4, -2, -2, 2, 6, 0, 2, 2, 2, 0, 8, 4, 0, 3, 3, 8, 2

Offset: 0

Michael Somos, Sep 21 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Zagier (2009) refers to Case D corresponding to the Apery numbers (A005258).

			G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A005258, A078905, A229802.

A := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* Michael Somos, Jun 10 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ Re[(3 - I) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, real( (3 - I) * [ 0, 1, I, -I, -1][ d%5 + 1])))};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -3, 2, 2, -3][k%5 + 1], 1 + x * O(x^n)), n))};

A = ModularForms( Gamma1(5), 1, prec=20) . basis(); A[0] + 3*A[1]; # Michael Somos, Jun 10 2014

Expansion of f(-q)^2 * (f(-q^5) / f(-q, -q^4))^5 = f(-q^2, -q^3)^2 * (f(-q^5) / f(-q, -q^4))^3 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 5 sequence [ 3, -2, -2, 3, -2, ...].
Moebius transform is period 5 sequence [ 3, 1, -1, -3, 0, ...]. - Michael Somos, Jun 10 2014
G.f. = g(t(q)) where g(), t() are the g.f. for A005258 and A078905.
G.f.: (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)))^5.

A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!Sum_{k = 0..n+1} (-1)^kC(n+1,k)(2n-k)! for n >= 1.

Original entry on oeis.org

-1, 1, 11, 181, 3539, 81901, 2203319, 67741129, 2346167879, 90449857081, 3843107102339, 178468044946621, 8994348275804891, 488964835817842021, 28523735794360301039, 1777328098986754744081, 117817961601577138782479, 8279178465722546926265329

Offset: 0

Peter Bala, Aug 14 2008

This sequence satisfies the recursion (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1) *(2*n^2 - 2*n+1)*a(n-1), which leads to a rapidly converging series for Napier's constant: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)* a(n-1)).

Notice the striking parallels with the theory of the Apéry numbers A(n) = A005258(n), which satisfy a similar recurrence relation n^2*A(n) - (n-1)^2*A(n-2) = (11*n^2-11*n+3)*A(n-1) and which appear in the 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) + ...].

Seiichi Manyama, Table of n, a(n) for n = 0..365
A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report., Math. Intelligencer 1 (1978/79), no 4, 195-203.

Cf. A005258, A060475, A086764, A143414, A143415.

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.)

a := n -> 1/(n-1)!*add((-1)^k*binomial(n+1,k)*(2*n-k)!, k = 0..n+1):
seq(a(n), n = 1..19);
# Alternative
a := n -> `if`(n<2, 2*n-1, (2*n)!/(n-1)!*hypergeom([-n-1], [-2*n], -1)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Nov 14 2018

Join[{-1}, Table[(1/(n-1)!)*Sum[(-1)^k*Binomial[n+1,k]*(2*n-k)!, {k, 0, n+1}], {n, 1, 50}]] (* G. C. Greubel, Oct 24 2017 *)

concat([-1], for(n=1,25, print1((1/(n-1)!)*sum(k=0,n+1, (-1)^k*binomial(n+1,k)*(2*n-k)!), ", "))) \\ G. C. Greubel, Oct 24 2017

a(0):= -1, a(n) = 1/(n-1)!*sum {k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.
Apart from the initial term, this sequence is the second superdiagonal of the square array A060475; equivalently, the second subdiagonal of the square array A086764.
Recurrence relation: a(0) = -1, a(1) = 1, (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), n >= 2.
Let b(n) denote the solution to this recurrence with initial conditions b(0) = 0, b(1) = 2. Then b(n) = A143414(n) = 1/(n-1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!. The rational number b(n)/a(n) is equal to the Padé approximation to exp(x) of degree (n-1,n+1) evaluated at x = 1 and b(n)/a(n) -> e very rapidly.
For example, b(100)/a(100) - e is approximately 1.934 * 10^(-436). The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^n *2*n^2 leads to rapidly converging series for e and 1/e: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)*a(n-1)) = 2*[1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...]; 1/e = 1/2 - 2*Sum_{n >= 2} (-1)^n * n^2/(b(n)*b(n-1)) = 1/2 - 2*[2^2/(2*30) - 3^2/(30*492) + 4^2/(492*9620) - ...].
Conjectural congruences: for r >= 0 and odd prime p, calculation suggests that a(p^r*(p+1)) + a(p^r) == 0 (mod p^(r+1)).
a(n) = ((2*n)!/(n-1)!)*hypergeom([-n-1], [-2*n], -1) for n >= 2. - Peter Luschny, Nov 14 2018
a(n) ~ 2^(2*n + 1/2) * n^(n+1) / exp(n + 1/2). - Vaclav Kotesovec, Jul 11 2021

cp's OEIS Frontend

A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.

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A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2*C(n,l)*C(k,l)*C(k+l,n)).

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A143003 a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+5)*a(n) - n^6*a(n-1).

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A183204 Central terms of triangle A181544.

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A036917 G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.

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A005802 Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e., 1234-avoiding permutations); vexillary permutations (i.e., 2143-avoiding).

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A143583 Apéry-like numbers: a(n) = (1/C(2n,n))*Sum_{k=0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).

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A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2C(n,l)C(k,l)*C(k+l,n)).

A143003 a(0) = 0, a(1) = 1, a(n+1) = (2n+1)(n^2+n+5)a(n) - n^6a(n-1).

A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.

A143583 Apéry-like numbers: a(n) = (1/C(2n,n))Sum_{k=0..n} C(2k,k)C(4k,2k)C(2n-2k,n-k)C(4n-4k,2n-2k).

A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!Sum_{k = 0..n+1} (-1)^kC(n+1,k)(2n-k)! for n >= 1.