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%I #96 Apr 05 2024 11:07:04
%S 1,6,42,312,2394,18756,149136,1199232,9729882,79527084,654089292,
%T 5408896752,44941609584,375002110944,3141107339328,26402533581312,
%U 222635989516122,1882882811380284,15967419789558804,135752058036988848,1156869080242393644
%N (n+1)^2*a(n+1) = (17n^2+17n+6)*a(n) - 72*n^2*a(n-1).
%C This is the Taylor expansion of a special point on a curve described by Beauville.
%C This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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="/A093388/b093388.txt">Table of n, a(n) for n = 0..1050</a> (terms 0..200 from Vincenzo Librandi)
%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.
%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="https://www.carmamaths.org/resources/jon/walks.pdf">Some Arithmetic Properties of Short Random Walk Integrals</a>, May 2011.
%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 Matthijs Coster, <a href="http://www.coster.demon.nl/sequences.htm">Sequences</a>
%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 F 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 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%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 Armin Straub, <a href="http://arminstraub.com/pub/dissertation">Arithmetic aspects of random walks and methods in definite integration</a>, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012
%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%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 H. Verrill, <a href="http://www.mpim-bonn.mpg.de/preprints?title=%22Some+congruences+related+to+modular+forms%22">Some congruences related to modular forms</a>, Section 2.2.
%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 F in sporadic solutions table of page 5.
%F a(n) = (-1)^n * Sum_{k=0..n} binomial(n, k) * (-8)^k * Sum_{j=0..n-k} binomial(n-k, j)^3. - Helena Verrill (verrill(AT)math.lsu.edu), Aug 09 2004
%F G.f.: hypergeom([1/3, 2/3],[1],x^2*(8*x-1)/(2*x-1/3)^3)/(1-6*x). - _Mark van Hoeij_, Oct 25 2011
%F a(n) ~ 3^(2*n+3/2)/(Pi*n). - _Vaclav Kotesovec_, Oct 14 2012
%F G.f. A(x) satisfies: 0 = x*(x+8)*(x+9)*y'' + (3*x^2 + 34*x + 72)*y' + (x+6)*y, where y(x) = A(-x/72). - _Gheorghe Coserea_, Aug 26 2016
%F From _Bradley Klee_, Jun 05 2023: (Start)
%F The g.f. T(x) obeys a period-annihilating ODE:
%F 0=6*(-1 + 12*x)*T(x) + (1 - 34*x + 216*x^2)*T'(x) + x*(-1 + 8*x)*(-1 + 9*x)*T''(x).
%F The periods ODE can be derived from the following Weierstrass data:
%F g2 = 12*(-1 + 6*x)*(-1 + 18*x - 84*x^2 + 24*x^3);
%F g3 = -8*(1 - 12*x + 24*x^2)*(-1 + 24*x - 192*x^2 + 504*x^3 + 72*x^4);
%F which determine an elliptic surface with four singular fibers. (End)
%e A(x) = 1 + 6*x + 42*x^2 + 312*x^3 + 2394*x^4 + 18756*x^5 + ... is the g.f.
%p f:=proc(n) option remember; local m; if n=0 then RETURN(1); fi; if n=1 then RETURN(6); fi; m:=n-1; ((17*m^2+17*m+6)*f(n-1)-72*m^2*f(n-2))/n^2; end;
%t Table[(-1)^n*Sum[Binomial[n,k]*(-8)^k*Sum[Binomial[n-k,j]^3,{j,0,n-k}],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o (PARI) a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*(-8)^k*sum(j=0,n-k,binomial(n-k,j)^3));
%o (PARI)
%o seq(N) = {
%o my(a = vector(N)); a[1] = 6; a[2] = 42;
%o for (n=3, N, a[n] = ((17*n^2 - 17*n + 6)*a[n-1] - 72*(n-1)^2*a[n-2])/n^2);
%o concat(1,a);
%o };
%o seq(20) \\ _Gheorghe Coserea_, Aug 26 2016
%Y This is the seventh sequence in the family beginning A002894, A006077, A081085, A005258, A000172, A002893.
%Y Cf. 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
%O 0,2
%A _Matthijs Coster_, Apr 29 2004
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