A033686 - OEIS

%I #58 Dec 17 2022 02:16:04

%S 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,

%T 32,62,28,44,30,56,40,57,34,70,36,72,38,70,48,62,52,85,44,68,46,112,

%U 56,74,50,100,64,80,64,98,56,108,58,124,60,112,76,112,64,98,66,155,80,104

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

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

%C Number of partition pairs of n where each partition is 3-core (see Theorem 2.1 of Wang link). - _Michel Marcus_, Jul 14 2015

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

%H Muniru A Asiru, <a href="/A033686/b033686.txt">Table of n, a(n) for n = 0..20000</a>

%H Kevin B. Ford, <a href="https://doi.org/10.2307/2160547">Some infinite series identities</a>, Proc. Amer. Math. Soc., Volume 119, Number 3 (November 1993), 1019-1020.

%H Liuquan Wang, <a href="http://arxiv.org/abs/1507.03099">Explicit Formulas for Partition Pairs and Triples with 3-Cores</a>, arXiv:1507.03099 [math.NT], 2015.

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

%F 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

%F Expansion of q^(-2/3) * (eta(q^3)^3 / eta(q))^2 in powers of q. - _Michael Somos_, Mar 16 2012

%F Convolution square of A033687. - _Michael Somos_, Oct 17 2006

%F 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.

%F 27 * a(n) = A096726(3*n + 2) - A281722(3*n + 2). - _Michael Somos_, Sep 04 2017

%F a(n) = A144615(n)/3. - _Robert G. Wilson v_, Jan 12 2018

%F From _Peter Bala_, Jan 07 2021: (Start)

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

%F 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).

%F 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)

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

%e 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 + ...

%e 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 + ...

%e 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) + ...

%p with(numtheory): seq(sigma(3*n-1)/3, n=1..2000); # _Muniru A Asiru_, Jan 18 2018

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* _Michael Somos_, May 26 2014 *)

%t Array[ DivisorSigma[1, 3 # - 1]/3 &, 69] (* _Robert G. Wilson v_, Jan 12 2018 *)

%o (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 */

%o (PARI) a(n)=sigma(3*n+2)/3; \\ _Michel Marcus_, Jul 14 2015

%o (Sage) ModularForms( Gamma0(9), 2, prec=195).2 # _Michael Somos_, May 26 2014

%o (Magma) Basis( ModularForms( Gamma0(9), 2), 195)[3]; /* _Michael Somos_, Jul 14 2015 */

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

%o (Magma) [SumOfDivisors(3*n+2)/3: n in [0..70]]; // _Vincenzo Librandi_, Jan 13 2018

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_