A006077 - OEIS

%I M2775 #140 Jun 15 2023 06:30:31

%S 1,3,9,21,9,-297,-2421,-12933,-52407,-145293,-35091,2954097,25228971,

%T 142080669,602217261,1724917221,283305033,-38852066421,-337425235479,

%U -1938308236731,-8364863310291,-24286959061533,-3011589296289,574023003011199,5028616107443691

%N (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 This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004

%C 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 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 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017

%C 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%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D 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 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 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 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 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</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 B 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 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-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 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 G.f.: hypergeom([1/3, 2/3], [1], x^3/(x-1/3)^3) / (1-3*x). - _Mark van Hoeij_, Oct 25 2011

%F 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 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 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1). - _Peter Luschny_, Nov 01 2017

%F From _Bradley Klee_, Jun 05 2023: (Start)

%F The g.f. T(x) obeys a period-annihilating ODE:

%F 0=3*(-1 + 9*x)*T(x) + (-1 + 9*x)^2*T'(x) +  x*(1 - 9*x + 27*x^2)*T''(x).

%F The periods ODE can be derived from the following Weierstrass data:

%F g2 = 3*(-8 + 9*(1 - 9*x)^3)*(1 - 9*x);

%F g3 = 8 - 36*(1 - 9*x)^3 + 27*(1 - 9*x)^6;

%F which determine an elliptic surface with four singular fibers. (End)

%e G.f. = 1 + 3*x + 9*x^2 + 21*x^3 + 9*x^4 - 297*x^5 - 2421*x^6 - 12933*x^7 - ...

%p a := n -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1):

%p seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Nov 01 2017

%t 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 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 (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 (PARI)

%o diag(expr, N=22, var=variables(expr)) = {

%o   my(a = vector(N));

%o   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));

%o   for (n = 1, N, a[n] = expr;

%o     for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));

%o   return(a);

%o };

%o diag(1/(1 + x^3 + y^3 + z^3 - 3*x*y*z), 25)

%o (PARI)

%o seq(N) = {

%o   my(a = vector(N)); a[1] = 3; a[2] = 9;

%o   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   concat(1,a);

%o };

%o seq(24)

%o \\ 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 \\ _Gheorghe Coserea_, Nov 09 2017

%o (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 Related to diagonal of rational functions: A268545-A268555.

%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 sign

%O 0,2

%A _N. J. A. Sloane_

%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000