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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)
4
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

REFERENCES

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

T. D. Noe, Table of n, a(n) for n = 0..200

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.

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

It is known that 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

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.

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 July 26 23:50 EDT 2016. Contains 275061 sequences.