A008654 -id:A008654 - cp's OEIS frontend

A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

1, -9, 27, -9, -117, 216, 27, -450, 459, -9, -648, 1080, -117, -1530, 1350, 216, -1845, 2592, 27, -3258, 2808, -450, -3240, 4752, 459, -5409, 4590, -9, -5850, 7560, -648, -8658, 7371, 1080, -7776, 10800, -117, -12330, 9774, -1530, -11016, 15120, 1350, -16650

Offset: 0

Michael Somos, Jun 17 2005

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Number 4 of the 74 eta-quotients listed in Table I of Martin (1996).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 313, Equ. (14.2.13).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
George E. Andrews and Bruce C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
Jonathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. See p. 697.
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A005928, A008654, A103440, A106402, A134340.

A := Basis( ModularForms( Gamma1(3), 3), 44); A[1] - 9*A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, Boole[ n == 0], - 9 DivisorSum[ n, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)

{a(n) = if( n<1, n==0, -9 * sumdiv( n, d, d^2 * kronecker(-3, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^9 / eta(x^3 + A)^3, n))};

G.f.: Product_{k>0} (1 - x^k)^9 / (1 - x^3)^3 = 1 - 9 * Sum_{k>0} x^k * (1 - x^k -6 * x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3.
Expansion of b(q)^3 in powers of q where b() is a cubic AGM theta function.
Euler transform of period 3 sequence [ -9, -9, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u*w * (u + 6*v - 8*w).
Given A = A0 + A1 + A2 is the 3-section, then 0 = A1^3 + A2^3 - 3*A0*A1*A2. A0 = A(q^3) = b(q^3)^3, A1 = -3 * a(q^3)^2 * c(q^3), A2 = 3 * a(q^3) * c(q^3)^2 where a(), b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 19683^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106402. - Michael Somos, Mar 11 2012
a(n) = -9 * A103440(n) unless n = 0. a(6*n + 5) = 216 * A134340(n).
A008654(n) = a(n) + 27 * A106402(n) is the identity a(q)^3 = b(q)^3 + c(q)^3. - Michael Somos, Jul 19 2012
a(n) = -9 * b(n) where b(n) is multiplicative with a(0) = 1, b(p^e) = 1, if p=3, b(p^e) = b(p) * b(p^(e-1)) + Kronecker(-3, p) * p^2 * b(p^(e-2)) otherwise. - Michael Somos, May 18 2015
Convolution cube of A005928. - Michael Somos, May 18 2015

A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

Original entry on oeis.org

1, 24, 216, 888, 1752, 3024, 7992, 8256, 14040, 24216, 27216, 31968, 64824, 52752, 74304, 111888, 112344, 117936, 217944, 164640, 220752, 305472, 287712, 292032, 519480, 378024, 474768, 654072

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Convolution of A008654 and A004016. Convolution square of A008653. - R. J. Mathar, Feb 22 2021

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

A008655 := proc(n)
        add( A004016(i)*x^i,i=0..n) ;
        coeftayl(%^4,x=0,n) ;
end proc: # R. J. Mathar, Feb 22 2021

terms = 28; s = (EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(81*EllipticTheta[3, 0, q^3]^4) + O[q]^(2 terms); CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 07 2017, from LatticeData(A2) *)

Expansion of (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^4.

A320676 Expansion of (r(q) * s(q))^3 where r(), s() are cubic AGM theta functions.

Original entry on oeis.org

1, 9, -27, -261, 765, 2214, -11529, 11304, 24813, -81423, 71118, 106812, -354609, 262350, 385992, -1049166, 739917, 990306, -2713203, 1709604, 2287710, -5646600, 3707532, 4448952, -11344833, 6737319, 8450838, -19943757, 12298248, 14238558, -34639974, 19856736

Offset: 0

Seiichi Manyama, Oct 19 2018

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Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004010, A008654, A281722.

Expansion of (eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2)^3 in powers of q.

cp's OEIS Frontend

A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

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A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

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A320676 Expansion of (r(q) * s(q))^3 where r(), s() are cubic AGM theta functions.

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