

A006077


(n+1)^2*a(n+1) = (9n^2+9n+3)*a(n)  27*n^2*a(n1), 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
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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) Hankeltype 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.  ZhiWei 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(p1) == (1)^{(p+1)/2}*(xp/x) (mod p^2), where W(n) is defined as the above.  ZhiWei Sun, Aug 23 2013


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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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).
Z.W. Sun, Connnections between p = x^2+3*y^2 and Franel numbers, J. Number Theory 133(2013), 29142928.
D. Zagier, Integral solutions of Aperylike recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349366.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..200
ZhiWei Sun, Three mysterious conjectures on Hankeltype determinants, a message to Number Theory List, August 22, 2013.


FORMULA

G.f.: hypergeom([1/3, 2/3],[1],x^3/(x1/3)^3)/(13*x).  Mark van Hoeij, Oct 25 2011
It is known that a(n) = sum_{k=0}^{[n/3]}(1)^k*3^(n3k)*C(n,3k)*C(2k,k)*C(3k,k).  ZhiWei 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


CROSSREFS

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



