A281722 -id:A281722 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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) + ...

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

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.

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

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

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

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

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

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

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 *)

{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 */

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

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

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

A096726 Expansion of eta(q^3)^10 / (eta(q) * eta(q^9))^3 in powers of q.

Original entry on oeis.org

1, 3, 9, 12, 21, 18, 36, 24, 45, 12, 54, 36, 84, 42, 72, 72, 93, 54, 36, 60, 126, 96, 108, 72, 180, 93, 126, 12, 168, 90, 216, 96, 189, 144, 162, 144, 84, 114, 180, 168, 270, 126, 288, 132, 252, 72, 216, 144, 372, 171, 279, 216, 294, 162, 36, 216, 360, 240, 270, 180, 504

Offset: 0

Michael Somos, Jul 06 2004

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

			G.f. = 1 + 3*x + 9*x^2 + 12*x^3 + 21*x^4 + 18*x^5 + 36*x^6 + 24*x^7 + 45*x^8 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 475, Entry 7(i).

Antti Karttunen, Table of n, a(n) for n = 0..16384
Bruce C. Berndt, Song Heng Chan, Zhi-Guo Liu, and Hamza Yesilyurt, A new identity for (q;q)10 [inf] with an application to Ramanujan's partition congruence modulo 11, Quart. J. of Math., 55 (2004), pp. 13-30; alternative link.
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012).

Cf. A033686, A116607, A281722.

Cf. A004016, A005882, A005928.

A := Basis( ModularForms( Gamma0(9), 2), 61); A[1] + 3*A[2] + 9*A[3]; /* Michael Somos, Aug 25 2014 */

CoefficientList[ Series[1 + Sum[k(3x^k/(1 - x^k) - 27x^(9k)/(1 - x^(9k))), {k, 1, 60}], {x, 0, 60}], x] (* Robert G. Wilson v, Jul 14 2004 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ If[ Mod[ d, 9] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Aug 25 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^10 / (QPochhammer[ q] QPochhammer[ q^9])^3, {q, 0, n}]; (* Michael Somos, Aug 25 2014 *)

{a(n) = if( n<1, n==0, 3 * sigma(n) - if( n%9==0, 27 * sigma(n/9)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^10 / (eta(x + A) * eta(x^9 + A))^3, n))};

{a(n) = polcoeff( sum(k=1, n, k*3* (x^k / (1 - x^k) - 9*x^(9*k) / (1 - x^(9*k))), 1 + x * O(x^n)), n)};

G.f.: Product_{k>0} (1 - x^(3*k))^10 / ((1 - x^k) * (1 - x^(9*k)))^3 = 1 + Sum_{k>0} k * (3*x^k / (1 - x^k) - 27 * x^(9*k) / (1 - x^(9*k))).
Euler transform of period 9 sequence [ 3, 3, -7, 3, 3, -7, 3, 3, -4, ...].
a(n) = 3 * b(n) where b(n) is multiplicative and b(3^e) = 1 + 3*(e>0), b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w + 4*u*w^2 + v^3 - 6*u*v*w.
Expansion of b(q^3)^3 / b(q) = c(q)^3 / (9*c(q^3)) = (a(q)^2 + 3*a(q^3)^2) / 4 = (a(q)^2 + a(q)*b(q) + b(q)^2) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 9 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 25 2014
a(3*n + 2) = A281722(3*n + 2) + 27 * A033686(n). a(n) == A281722(n) (mod 27). - Michael Somos, Sep 04 2017
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/9 = 2.193245... . - Amiram Eldar, Dec 28 2023

A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.

Original entry on oeis.org

1, -3, 0, 3, 6, 0, -18, 3, 0, 12, 0, 0, 21, -15, 0, -36, -12, 0, 36, 21, 0, 24, 0, 0, -90, 15, 0, 12, -6, 0, 54, 12, 0, -72, 0, 0, 84, -33, 0, 42, 0, 0, -144, -24, 0, 72, 0, 0, 93, 18, 0, -108, 30, 0, 36, 0, 0, 60, 0, 0, -252, 3, 0, 96, 24, 0, 108, -15, 0

Offset: 0

Michael Somos, Feb 19 2017

sign

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (9*t)) = 729 (t/i)^2 g(t) where g() is the g.f. for A282611.

			G.f. = 1 - 3*q + 3*q^3 + 6*q^4 - 18*q^6 + 3*q^7 + 12*q^9 + 21*q^12 - 15*q^13 + ...

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

Cf. A030206, A281722, A282611.

A := Basis( ModularForms( Gamma0(27), 2), 69); A[1] - 3*A[2] + 3*A[4] + 6*A[5] - 18*A[6];

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^3]^2 / QPochhammer[ q^9], {q, 0, n}];

first(n)=my(q='x+O('x^(n+1))); Vec(eta(q)^3 * eta(q^3)^2 / eta(q^9)) \\ Charles R Greathouse IV, Jun 02 2017

Expansion of eta(q)^3 * eta(q^3)^2 / eta(q^9) in powers of q.
Euler transform of period 9 sequence [-3, -3, -5, -3, -3, -5, -3, -3, -4, ...].
a(3*n) = A281722(n). a(3*n + 1) = -3 * A030206(n). a(3*n + 2) = 0.

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

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A096726 Expansion of eta(q^3)^10 / (eta(q) * eta(q^9))^3 in powers of q.

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A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.

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

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