|
%I M2281 #54 Dec 17 2022 03:11:51
%S 3,3,6,0,6,3,6,0,3,6,6,0,6,0,6,0,9,6,0,0,6,3,6,0,6,6,6,0,0,0,12,0,6,3,
%T 6,0,6,6,0,0,3,6,6,0,12,0,6,0,0,6,6,0,6,0,6,0,9,6,6,0,6,0,0,0,6,9,6,0,
%U 0,6,6,0,12,0,6,0,6,0,0,0,6,6,12,0,0,3,12,0,0,6,6,0,6,0,6,0,3,6,0,0,12
%N Theta series of planar hexagonal lattice (A2) with respect to deep hole.
%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C On page 111 of Conway and Sloane is "If the origin is moved to a deep hole the theta series is Theta_{hex+[1]}(z) = theta_2(z) psi_6(3z) + theta_3(z) psi_3(3z) = 3 q^{1/3} + 3 q^{4/3} + 6 q^{7/3} + 6 q^{13/3} + ... (63)" where the psi_k() for integer k is defined on page 103 equation (11) as psi_k(z) = e^{Pi i/z^2} theta_3(Pi z/k | z) = Sum_{m in Z} q^{(m + 1/k)^2}. - _Michael Somos_, Sep 10 2018
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A005882/b005882.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H N. J. A. Sloane and B. K. Teo, <a href="http://dx.doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/3) * 3 * eta(q^3)^3 / eta(q) in powers of q.
%F Expansion of q^(-1/3) * c(q) in powers of q where c(q) is the third cubic AGM theta function.
%F Given g.f. A(x), then B(x) = x*A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 2*u*w^2 - u^2*w. - _Michael Somos_, Aug 15 2006
%F G.f.: 3 Product_{k>0} (1-q^(3k))^3/(1-q^k).
%F G.f.: Sum_{u,v in Z} x^(u*u + u*v + v*v + u + v). - _Michael Somos_, Jul 19 2014
%F a(n) = 3 * A033687(n). a(n) = A113062(3*n + 1) = A033685(3*n + 1).
%F Expansion of 2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - _Michael Somos_, Sep 07 2018
%F Sum_{k=1..n} a(k) ~ 2*Pi*n/sqrt(3). - _Vaclav Kotesovec_, Dec 17 2022
%e G.f. = 3 + 3*x + 6*x^2 + 6*x^4 + 3*x^5 + 6*x^6 + 3*x^8 + 6*x^9 + 6*x^10 + ...
%e G.f. = 3*q + 3*q^4 + 6*q^7 + 6*q^13 + 3*q^16 + 6*q^19 + 3*q^25 + 6*q^28 + ...
%t a[ n_] := SeriesCoefficient[ 3 QPochhammer[ q^3]^3 / QPochhammer[ q], {q, 0, n}]; (* _Michael Somos_, Jul 19 2014 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^3 + A)^3 / eta(x + A), n))}; /* _Michael Somos_, Aug 15 2006 */
%o (Magma) Basis( ModularForms( Gamma1(9), 1), 302)[2] * 3; /* _Michael Somos_, Jul 19 2014 */
%Y Essentially same as A033685 and A033687.
%Y Cf. A113062, A273845.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
|
