A081085 Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.
1, 4, 20, 112, 676, 4304, 28496, 194240, 1353508, 9593104, 68906320, 500281280, 3664176400, 27033720640, 200683238720, 1497639994112, 11227634469668, 84509490017680, 638344820152784, 4836914483890112, 36753795855173776, 279985580271435584, 2137790149251471680
Offset: 0
Examples
G.f. = A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + ...
References
- Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 0..200 from Vincenzo Librandi)
- 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.
- Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, page 657.
- Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
- E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
- Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See E p. 2.
- S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
- Xiao-Juan Ji and Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015.
- Bradley Klee, Checking Weierstrass data, 2023.
- Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
- 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.
- D. Zagier, Integral solutions of Apery-like recurrence equations. See line E in sporadic solutions table of page 5.
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.)
Programs
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Maple
seq(simplify(binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1)), n = 0..22); # Peter Bala, Jul 25 2024
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Mathematica
Table[Sum[Binomial[n,k]*Binomial[2*n-2*k,n-k]*Binomial[2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *) a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1/2, 1, 16 x (1 - 4 x)], {x, 0, n}]; (* Michael Somos, Oct 25 2014 *) a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / NestWhile[ {(#[[1]] + #[[2]])/2, Sqrt[#[[1]] #[[2]]]} &, {1, Series[ 1 - 8 x, {x, 0, n}]}, #[[1]] =!= #[[2]] &] // First, {x, 0, n}]]; (* Michael Somos, Oct 27 2014 *) CoefficientList[Series[2*EllipticK[1/(1 - 1/(4*x))^2] / (Pi*(1 - 4*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *) a[n_] := Binomial[2 n, n] HypergeometricPFQ[{1/2, -n, -n},{1, 1/2 - n}, -1]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 05 2022 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))}; -
PARI
{a(n) = if( n<0,0, 4^n * sum( k=0, n\2, binomial( n, 2*k) * binomial( 2*k, k)^2 / 16^k))}; -
PARI
{a(n)=n!*polcoeff(sum(k=0,n,(2*k)!*x^k/(k!)^3 +x*O(x^n))^2,n)} /* Paul D. Hanna, Sep 04 2009 */ -
Python
from math import comb def A081085(n): return sum((1<<(n-(m:=k<<1)<<1))*comb(n,m)*comb(m,k)**2 for k in range((n>>1)+1)) # Chai Wah Wu, Jul 09 2023
Formula
G.f.: 1 / AGM(1, 1 - 8*x).
E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) = binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1). - Vladeta Jovovic, Sep 16 2003
D-finite with recurrence (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1). - Matthijs Coster, Apr 28 2004
E.g.f.: [Sum_{n>=0} binomial(2n,n)*x^n/n! ]^2. - Paul D. Hanna, Sep 04 2009
G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2). - Mark van Hoeij, Oct 24 2011
a(n) = 2^(-n) * A053175(n).
a(n) ~ 2^(3*n+1)/(Pi*n). - Vaclav Kotesovec, Oct 13 2012
0 = x*(x+4)*(x+8)*y'' + (3*x^2 + 24*x + 32)*y' + (x+4)*y, where y(x) = A(x/-32). - Gheorghe Coserea, Aug 26 2016
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k)^2. - Seiichi Manyama, Apr 02 2017
a(n) = (1/Pi)^2*Integral_{0 <= x, y <= Pi} (4*cos(x)^2 + 4*cos(y)^2)^n dx dy. - Peter Bala, Feb 10 2022
a(n) = a(-1-n)*32^(n-1/2) and 0 = +a(n)*(+a(n+1)*(+32768*a(n+2) -23552*a(n+3) +3072*a(n+4)) +a(n+2)*(-8192*a(n+2) +8448*a(n+3) -1248*a(n+4)) +a(n+3)*(-512*a(n+3) +96*a(n+4))) +a(n+1)*(+a(n+1)*(-5120*a(n+2) +3840*a(n+3) -512*a(n+4)) +a(n+2)*(+1536*a(n+2) -1728*a(n+3) +264*a(n+4)) +a(n+3)*(+120*a(n+3) -23*a(n+4))) +a(n+2)*(+a(n+2)*(-32*a(n+2) +48*a(n+3) -8*a(n+4)) +a(n+3)*(-5*a(n+3) +a(n+4))) for all n in Z. - Michael Somos, Apr 04 2022
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=4*(-1 + 8*x)*T(x) + (1 - 24*x + 96*x^2)*T'(x) + x*(-1 + 4*x)*(-1 + 8*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 3*(1 - 16*(1 - 8*x)^2 + 16*(1 - 8*x)^4);
g3 = 1 + 30*(1 - 8*x)^2 - 96*(1 - 8*x)^4 + 64*(1 - 8*x)^6;
which determine an elliptic surface with four singular fibers. (End)
G.f.: Sum_{n>=0} binomial(2*n,n)^2 * x^n * (1 - 4*x)^n. - Paul D. Hanna, Apr 18 2024
From Peter Bala, Jul 25 2024: (Start)
a(n) = 2*Sum_{k = 1..n} (k/n)*binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) for n >= 1.
a(n-1) = (1/2)*Sum_{k = 1..n} (k/n)^2*binomial(n, k)*binomial(2*n-2*k, n-k)* binomial(2*k, k) for n >= 1. Cf. A002895. (End)
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