A033686 One-ninth of theta series of A2[hole]^2.

1, 2, 5, 4, 8, 6, 14, 8, 14, 10, 21, 16, 20, 14, 28, 16, 31, 18, 40, 20, 32, 28, 42, 24, 38, 32, 62, 28, 44, 30, 56, 40, 57, 34, 70, 36, 72, 38, 70, 48, 62, 52, 85, 44, 68, 46, 112, 56, 74, 50, 100, 64, 80, 64, 98, 56, 108, 58, 124, 60, 112, 76, 112, 64, 98, 66, 155, 80, 104

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Number of partition pairs of n where each partition is 3-core (see Theorem 2.1 of Wang link). - Michel Marcus, Jul 14 2015

Examples

			G.f. = 1 + 2*x + 5*x^2 + 4*x^3 + 8*x^4 + 6*x^5 + 14*x^6 + 8*x^7 + 14*x^8 + ...
G.f. = q^2 + 2*q^5 + 5*q^8 + 4*q^11 + 8*q^14 + 6*q^17 + 14*q^20 + 8*q^23 + ...
Theta series of A2[hole]^2 = c(q)^2 = 9*q^(2/3) + 18*q^(5/3) + 45*q^(8/3) + 36*q^(11/3) + 72*q^(14/3) + 54*q^(17/3) + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111, Eq (63)^2.

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..20000
• Kevin B. Ford, Some infinite series identities, Proc. Amer. Math. Soc., Volume 119, Number 3 (November 1993), 1019-1020.
• Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Crossrefs

Cf. A033687, A096726, A097723, A134079, A144615, A242874, A281722.

Cf. A000203 (sigma), A016789 (3n+2).

Programs

GAP

sequence := List([1..100010],n->Sigma(3*n-1)/3); # Muniru A Asiru, Dec 29 2017

Magma

Basis( ModularForms( Gamma0(9), 2), 195)[3]; /* Michael Somos, Jul 14 2015 */

Magma

[SumOfDivisors(3*n+2)/3: n in [0..70]]; // Vincenzo Librandi, Jan 13 2018

Maple

with(numtheory): seq(sigma(3*n-1)/3, n=1..2000); # Muniru A Asiru, Jan 18 2018

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 26 2014 *)
Array[ DivisorSigma[1, 3 # - 1]/3 &, 69] (* Robert G. Wilson v, Jan 12 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Oct 17 2006 */

PARI

a(n)=sigma(3*n+2)/3; \\ Michel Marcus, Jul 14 2015

Sage

ModularForms( Gamma0(9), 2, prec=195).2 # Michael Somos, May 26 2014

Formula

a(n) = sigma(3*n+2)/3. Euler transform of period 3 sequence [2, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004

Expansion of q^(-2/3) * c(q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function. - Michael Somos, Oct 17 2006

Expansion of q^(-2/3) * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Mar 16 2012

Convolution square of A033687. - Michael Somos, Oct 17 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A242874.

27 * a(n) = A096726(3*n + 2) - A281722(3*n + 2). - Michael Somos, Sep 04 2017

a(n) = A144615(n)/3. - Robert G. Wilson v, Jan 12 2018

From Peter Bala, Jan 07 2021: (Start)

a(n) = (-1)^n*A134079(n).

A(x) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(3*n+1))^2 = Sum_{n = -oo..oo} x^(4*n+2)/(1 - x^(3*n+2))^2 (apply Ford, equation 1, with c = x^(3/2), d = x^(1/2), |x| < 1 to the g.f. Sum_{n = -oo..oo} x^n /(1 - x^(3*n + 1)) of A033687).

Conjectural g.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(3*n+2))^2 = Sum_{n = -oo..oo} x^(5*n+1)/(1 - x^(3*n+1))^2. (End)

Sum_{k=1..n} a(k) = (2*Pi^2/27) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022