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A093388 (n+1)^2*a(n+1) = (17n^2+17n+6)*a(n) - 72*n^2*a(n-1).

%I A093388 #96 Apr 05 2024 11:07:04
%S A093388 1,6,42,312,2394,18756,149136,1199232,9729882,79527084,654089292,
%T A093388 5408896752,44941609584,375002110944,3141107339328,26402533581312,
%U A093388 222635989516122,1882882811380284,15967419789558804,135752058036988848,1156869080242393644
%N A093388 (n+1)^2*a(n+1) = (17n^2+17n+6)*a(n) - 72*n^2*a(n-1).
%C A093388 This is the Taylor expansion of a special point on a curve described by Beauville.
%C A093388 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%D A093388 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%H A093388 Seiichi Manyama, <a href="/A093388/b093388.txt">Table of n, a(n) for n = 0..1050</a> (terms 0..200 from Vincenzo Librandi)
%H A093388 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 A093388 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 A093388 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 A093388 Matthijs Coster, <a href="http://www.coster.demon.nl/sequences.htm">Sequences</a>
%H A093388 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 A093388 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 A093388 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H A093388 Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.
%H A093388 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 A093388 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 A093388 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A093388 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 A093388 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 A093388 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 A093388 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 A093388 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 A093388 a(n) ~ 3^(2*n+3/2)/(Pi*n). - _Vaclav Kotesovec_, Oct 14 2012
%F A093388 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 A093388 From _Bradley Klee_, Jun 05 2023: (Start)
%F A093388 The g.f. T(x) obeys a period-annihilating ODE:
%F A093388 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 A093388 The periods ODE can be derived from the following Weierstrass data:
%F A093388 g2 = 12*(-1 + 6*x)*(-1 + 18*x - 84*x^2 + 24*x^3);
%F A093388 g3 = -8*(1 - 12*x + 24*x^2)*(-1 + 24*x - 192*x^2 + 504*x^3 + 72*x^4);
%F A093388 which determine an elliptic surface with four singular fibers.  (End)
%e A093388 A(x) = 1 + 6*x + 42*x^2 + 312*x^3 + 2394*x^4 + 18756*x^5 + ... is the g.f.
%p A093388 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 A093388 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 A093388 (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 A093388 (PARI)
%o A093388 seq(N) = {
%o A093388   my(a = vector(N)); a[1] = 6; a[2] = 42;
%o A093388   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 A093388   concat(1,a);
%o A093388 };
%o A093388 seq(20)  \\ _Gheorghe Coserea_, Aug 26 2016
%Y A093388 This is the seventh sequence in the family beginning A002894, A006077, A081085, A005258, A000172, A002893.
%Y A093388 Cf. A091401.
%Y A093388 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 A093388 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 A093388 nonn
%O A093388 0,2
%A A093388 _Matthijs Coster_, Apr 29 2004