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A033686 One-ninth of theta series of A2[hole]^2. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

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

EXAMPLE

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) + ...

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 *)

PROG

(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

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

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

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

CROSSREFS

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

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

Sequence in context: A081556 A187012 A134079 * A243973 A286015 A183542

Adjacent sequences:  A033683 A033684 A033685 * A033687 A033688 A033689

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)