Voozh

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.

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

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

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

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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See B p. 2.

S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.

Bradley Klee, Checking Weierstrass data, 2023.

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

Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.

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

Zhi-Wei 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..floor(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

a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1). - Peter Luschny, Nov 01 2017

From Bradley Klee, Jun 05 2023: (Start)

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

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

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

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

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

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

EXAMPLE

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

MAPLE

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

seq(simplify(a(n)), n=0..24); # Peter Luschny, Nov 01 2017

MATHEMATICA

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 *)

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 *)

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 [for (-1)^n*a(n)]

(PARI)

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

my(a = vector(N));

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

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

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

return(a);

};

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

(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)

\\ 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

\\ Gheorghe Coserea, Nov 09 2017

(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 */

CROSSREFS

Related to diagonal of rational functions: A268545-A268555.

Cf. A091401.