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%I #146 Apr 18 2024 09:32:15
%S 1,4,20,112,676,4304,28496,194240,1353508,9593104,68906320,500281280,
%T 3664176400,27033720640,200683238720,1497639994112,11227634469668,
%U 84509490017680,638344820152784,4836914483890112,36753795855173776,279985580271435584,2137790149251471680
%N Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.
%C AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.
%C This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004
%C This is the exponential (also known as binomial) convolution of sequence A000984 (central binomial) with itself. See the V. Jovovic e.g.f. and a(n) formulas given below. - _Wolfdieter Lang_, Jan 13 2012
%C This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C The recursion (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1) with n=0 has zero coefficient for a(-1) and thus a(-1) is not determined uniquely by it, but defining a(-1) = 2^(-5/2) makes a(n) = a(-1-n) * 32^(n-1/2) true for all n in Z. - _Michael Somos_, Apr 05 2022
%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%H Seiichi Manyama, <a href="/A081085/b081085.txt">Table of n, a(n) for n = 0..1110</a> (terms 0..200 from Vincenzo Librandi)
%H B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.
%H Arnaud Beauville, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5543443c/f31.item">Les familles stables de courbes sur P_1 admettant quatre fibres singulières</a>, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, page 657.
%H Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023.
%H E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apery-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
%H Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See E p. 2.
%H S. Herfurtner, <a href="https://doi.org/10.1007/BF01445211">Elliptic surfaces with four singular fibres</a>, Mathematische Annalen, 1991. <a href="https://archive.mpim-bonn.mpg.de/id/eprint/860/">Preprint</a>.
%H Xiao-Juan Ji and Zhi-Hong Sun, <a href="http://arxiv.org/abs/1505.00668">Congruences for Catalan-Larcombe-French numbers</a>, arXiv:1505.00668 [math.NT], 2015.
%H Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.
%H Stéphane Ouvry and Alexios Polychronakos, <a href="https://arxiv.org/abs/2006.06445">Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers</a>, arXiv:2006.06445 [math-ph], 2020.
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020.
%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apery-like recurrence equations</a>. See line E in sporadic solutions table of page 5.
%F G.f.: 1 / AGM(1, 1 - 8*x).
%F E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - _Vladeta Jovovic_, Aug 20 2003
%F 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
%F 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
%F E.g.f.: [Sum_{n>=0} binomial(2n,n)*x^n/n! ]^2. - _Paul D. Hanna_, Sep 04 2009
%F G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2). - _Mark van Hoeij_, Oct 24 2011
%F a(n) = 2^(-n) * A053175(n).
%F a(n) ~ 2^(3*n+1)/(Pi*n). - _Vaclav Kotesovec_, Oct 13 2012
%F 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
%F 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
%F 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
%F 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
%F a(n) = binomial(2*n, n)*hypergeom([1/2, -n, -n], [1, 1/2 - n], -1). - _Peter Luschny_, Apr 05 2022
%F From _Bradley Klee_, Jun 05 2023: (Start)
%F The g.f. T(x) obeys a period-annihilating ODE:
%F 0=4*(-1 + 8*x)*T(x) + (1 - 24*x + 96*x^2)*T'(x) + x*(-1 + 4*x)*(-1 + 8*x)*T''(x).
%F The periods ODE can be derived from the following Weierstrass data:
%F g2 = 3*(1 - 16*(1 - 8*x)^2 + 16*(1 - 8*x)^4);
%F g3 = 1 + 30*(1 - 8*x)^2 - 96*(1 - 8*x)^4 + 64*(1 - 8*x)^6;
%F which determine an elliptic surface with four singular fibers. (End)
%F G.f.: Sum_{n>=0} binomial(2*n,n)^2 * x^n * (1 - 4*x)^n. - _Paul D. Hanna_, Apr 18 2024
%e G.f. = A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + ...
%t 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 *)
%t a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1/2, 1, 16 x (1 - 4 x)], {x, 0, n}]; (* _Michael Somos_, Oct 25 2014 *)
%t 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 *)
%t CoefficientList[Series[2*EllipticK[1/(1 - 1/(4*x))^2] / (Pi*(1 - 4*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jan 13 2019 *)
%t a[n_] := Binomial[2 n, n] HypergeometricPFQ[{1/2, -n, -n},{1, 1/2 - n}, -1];
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Apr 05 2022 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))};
%o (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))};
%o (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 */
%o (Python)
%o from math import comb
%o 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
%Y Cf. A053175, A089603, A091401.
%Y 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.)
%Y 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.
%K nonn,easy,changed
%O 0,2
%A _Michael Somos_, Mar 04 2003
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