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A006077 (n+1)^2*a(n+1) = (9n^2+9n+3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = 3.

%I A006077 M2775 #140 Jun 15 2023 06:30:31
%S A006077 1,3,9,21,9,-297,-2421,-12933,-52407,-145293,-35091,2954097,25228971,
%T A006077 142080669,602217261,1724917221,283305033,-38852066421,-337425235479,
%U A006077 -1938308236731,-8364863310291,-24286959061533,-3011589296289,574023003011199,5028616107443691
%N A006077 (n+1)^2*a(n+1) = (9n^2+9n+3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = 3.
%C A006077 This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004
%C A006077 Conjecture: Let W(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. If n == 1 (mod 3) then W(n) = 0. When n == 0 or 2 (mod 3), W(n)*(-1)^(floor((n+1)/3))/6^n is always a positive odd integer. - _Zhi-Wei Sun_, Aug 21 2013
%C A006077 Conjecture: Let p == 1 (mod 3) be a prime, and write 4*p = x^2 + 27*y^2 with x, y integers and x == 1 (mod 3). Then W(p-1) == (-1)^{(p+1)/2}*(x-p/x) (mod p^2), where W(n) is defined as the above. - _Zhi-Wei Sun_, Aug 23 2013
%C A006077 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A006077 Diagonal of rational functions 1/(1 - (x^2*y + y^2*z - z^2*x + 3*x*y*z)), 1/(1 - (x^3 + y^3 - z^3 + 3*x*y*z)), 1/(1 + x^3 + y^3 + z^3 - 3*x*y*z). - _Gheorghe Coserea_, Aug 04 2018
%D A006077 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%D A006077 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006077 D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.
%H A006077 Seiichi Manyama, <a href="/A006077/b006077.txt">Table of n, a(n) for n = 0..1400</a> (terms 0..200 from T. D. Noe)
%H A006077 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 A006077 A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
%H A006077 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A006077 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 B p. 2.
%H A006077 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 A006077 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H A006077 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 A006077 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 A006077 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;fb189464.1308">Three mysterious conjectures on Hankel-type determinants</a>, a message to Number Theory List, August 22, 2013.
%H A006077 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/150f.pdf">Connections between p = x^2+3*y^2 and Franel numbers</a>, J. Number Theory 133(2013), 2914-2928.
%F A006077 G.f.: hypergeom([1/3, 2/3], [1], x^3/(x-1/3)^3) / (1-3*x). - _Mark van Hoeij_, Oct 25 2011
%F A006077 a(n) = Sum_{k=0..floor(n/3)}(-1)^k*3^(n-3k)*C(n,3k)*C(2k,k)*C(3k,k). - _Zhi-Wei Sun_, Aug 21 2013
%F A006077 0 = x*(x^2+9*x+27)*y'' + (3*x^2 + 18*x + 27)*y' + (x + 3)*y, where y(x) = A(x/-27). - _Gheorghe Coserea_, Aug 26 2016
%F A006077 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1). - _Peter Luschny_, Nov 01 2017
%F A006077 From _Bradley Klee_, Jun 05 2023: (Start)
%F A006077 The g.f. T(x) obeys a period-annihilating ODE:
%F A006077 0=3*(-1 + 9*x)*T(x) + (-1 + 9*x)^2*T'(x) +  x*(1 - 9*x + 27*x^2)*T''(x).
%F A006077 The periods ODE can be derived from the following Weierstrass data:
%F A006077 g2 = 3*(-8 + 9*(1 - 9*x)^3)*(1 - 9*x);
%F A006077 g3 = 8 - 36*(1 - 9*x)^3 + 27*(1 - 9*x)^6;
%F A006077 which determine an elliptic surface with four singular fibers. (End)
%e A006077 G.f. = 1 + 3*x + 9*x^2 + 21*x^3 + 9*x^4 - 297*x^5 - 2421*x^6 - 12933*x^7 - ...
%p A006077 a := n -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1):
%p A006077 seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Nov 01 2017
%t A006077 Table[Sum[(-1)^k*3^(n - 3*k)*Binomial[n, 3*k]*Binomial[2*k, k]* Binomial[3*k, k], {k, 0, Floor[n/3]}], {n, 0, 50}] (* _G. C. Greubel_, Oct 24 2017 *)
%t A006077 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/3, 2/3}, {1}, x^3 / (x - 1/3)^3 ] / (1 - 3 x), {x, 0, n}]; (* _Michael Somos_, Nov 01 2017 *)
%o A006077 (PARI) subst(eta(q)^3/eta(q^3), q, serreverse(eta(q^9)^3/eta(q)^3*q)) \\ (generating function) Helena Verrill (verrill(AT)math.lsu.edu), Apr 20 2009 [for (-1)^n*a(n)]
%o A006077 (PARI)
%o A006077 diag(expr, N=22, var=variables(expr)) = {
%o A006077   my(a = vector(N));
%o A006077   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
%o A006077   for (n = 1, N, a[n] = expr;
%o A006077     for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));
%o A006077   return(a);
%o A006077 };
%o A006077 diag(1/(1 + x^3 + y^3 + z^3 - 3*x*y*z), 25)
%o A006077 (PARI)
%o A006077 seq(N) = {
%o A006077   my(a = vector(N)); a[1] = 3; a[2] = 9;
%o A006077   for (n = 2, N-1, a[n+1] = ((9*n^2+9*n+3)*a[n] - 27*n^2*a[n-1])/(n+1)^2);
%o A006077   concat(1,a);
%o A006077 };
%o A006077 seq(24)
%o A006077 \\ test: y=subst(Ser(seq(202)), 'x, -'x/27); 0 == x*(x^2+9*x+27)*y'' + (3*x^2+18*x+27)*y' + (x+3)*y
%o A006077 \\ _Gheorghe Coserea_, Nov 09 2017
%o A006077 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^n * polcoeff( subst(eta(x + A)^3 / eta(x^3 + A), x, serreverse( x * eta(x^9 + A)^3 / eta(x + A)^3)), n))}; /* _Michael Somos_, Nov 01 2017 */
%Y A006077 Related to diagonal of rational functions: A268545-A268555.
%Y A006077 Cf. A091401.
%Y A006077 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 A006077 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 A006077 sign
%O A006077 0,2
%A A006077 _N. J. A. Sloane_
%E A006077 More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000