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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.
(Formerly M2775)
47
1, 3, 9, 21, 9, -297, -2421, -12933, -52407, -145293, -35091, 2954097, 25228971, 142080669, 602217261, 1724917221, 283305033, -38852066421, -337425235479, -1938308236731, -8364863310291, -24286959061533, -3011589296289, 574023003011199, 5028616107443691 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004

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

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

Diagonal of rational function R(x,y,z) = 1/(1 + x^3 + y^3 + z^3 - 3*x*y*z). - Gheorghe Coserea, Jul 01 2016

This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

REFERENCES

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

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

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.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1400 (terms 0..200 from T. D. Noe)

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

M. Coster, Email, Nov 1990

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

Zhi-Wei Sun, Three mysterious conjectures on Hankel-type determinants, a message to Number Theory List, August 22, 2013.

Z.-W. Sun, Connections between p = x^2+3*y^2 and Franel numbers, J. Number Theory 133(2013), 2914-2928.

FORMULA

G.f.: hypergeom([1/3, 2/3],[1],x^3/(x-1/3)^3)/(1-3*x). - Mark van Hoeij, Oct 25 2011

a(n) = Sum_{k=0..[n/3]}(-1)^k*3^(n-3k)*C(n,3k)*C(2k,k)*C(3k,k). - Zhi-Wei Sun, Aug 21 2013

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

EXAMPLE

A(x) = 1 + 3*x + 9*x^2 + 21*x^3 + 9*x^4 - 297*x^5 + ... is the g.f.

PROG

(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

(PARI)

my(x='x, y='y, z='z);

R = 1/(1 + x^3 + y^3 + z^3 - 3*x*y*z);

diag(n, expr, var) = {

  my(a = vector(n));

  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));

  for (k = 1, n, a[k] = expr;

       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));

  return(a);

};

diag(25, R, [x, y, z])

(PARI)

seq(N) = {

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

  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);

  concat(1, a);

};

seq(24)  \\ Gheorghe Coserea, Jul 01 2016

CROSSREFS

Related to diagonal of rational functions: A268545-A268555.

Cf. A091401.

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.)

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.

Sequence in context: A196212 A146219 A197403 * A109612 A032668 A050839

Adjacent sequences:  A006074 A006075 A006076 * A006078 A006079 A006080

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified October 22 03:43 EDT 2017. Contains 293756 sequences.