OFFSET
0,2
REFERENCES
M. Petkovsek et al., "A=B", Peters, p. ix of second printing.
LINKS
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.
FORMULA
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)
EXAMPLE
G.f. = 1 + 8*x + 88*x^2 + 1088*x^3 + 14296*x^5 + 195008*x^5 + ... - Michael Somos, May 29 2023
MATHEMATICA
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 *)
PROG
(Haskell)
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
(PARI) 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
(PARI) a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)); /* Michael Somos, May 29 2023 */
CROSSREFS
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.)
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
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
STATUS
approved