login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
5
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

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

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.

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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 3 20:40 EST 2016. Contains 278745 sequences.