A033686 -id:A033686 - cp's OEIS frontend

A098098 a(n) = sigma(6*n+5)/6.

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72

Offset: 0

Vladeta Jovovic, Sep 14 2004

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...

Ivan Neretin, Table of n, a(n) for n = 0..10000
Nayandeep Deka Baruah and Kallol Nath, Infinite families of arithmetic identities and congruences for bipartitions with 3-cores, Journal of Number Theory, Volume 149, April 2015, Pages 92-104.

Cf. A000203, A016969, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.

Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */

Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)

PARI
```
a(n) = sigma(6*n + 5)/6
```

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) =  A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A097723 One fourth of sum of divisors of 4n+3.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 10, 8, 12, 14, 11, 12, 18, 18, 15, 26, 17, 18, 31, 20, 21, 30, 28, 30, 39, 26, 27, 38, 36, 36, 42, 32, 33, 60, 35, 42, 57, 38, 48, 54, 41, 42, 65, 62, 45, 62, 54, 48, 84, 50, 60, 78, 53, 66, 74, 56, 57, 96, 72, 60, 91, 70, 63, 108, 76, 66, 90, 68, 93, 104, 71, 84, 98

Offset: 0

N. J. A. Sloane, Sep 11 2004

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 2*x + 3*x^2 + 6*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 8*x^7 + 12*x^8 + ...
q^3 + 2*q^7 + 3*q^11 + 6*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 8*q^31 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 76, Eq. (31.54).

Ivan Neretin, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions.
Min Wang, Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Kenneth S. Williams, n = delta+delta+2(delta+delta), Far East J. Math. Sci. 11 (2003), 233-240.

Cf. A000203, A004767, A033686.

Table[DivisorSigma[1, 4n+3]/4, {n, 0, 72}] (* Jean-François Alcover, Nov 30 2015 *)

{a(n) = if( n<0, 0, sigma(4*n + 3) / 4)} /* Michael Somos, Jul 05 2006 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A))^2, n))} /* Michael Somos, Jul 05 2006 */

Euler transform of period 4 sequence [2, 0, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of q^(-3/4) * eta^2(q^2) * eta^4(q^4) / eta^2(q) in powers of q. - Michael Somos, Jul 05 2006
Expansion of q^(-3/2) * (theta_2(q) * theta_2(q^2))^2 / 16 in powers of q^2. - Michael Somos, Jul 05 2006
Expansion of (psi(x) * psi(x^2))^2 in powers of x where psi() is a Ramanujan theta function.
a(n) = sigma(4*n + 3) / 4 = A000203(4*n + 3) / 4.
a(n) = number of solutions of 8*n + 6 = x^2 + y^2 + 2*z^2 + 2*w^2 in positive odd integers.
a(n) = number of representations of n as the sum of two triangular numbers and twice two triangular numbers. - Michael Somos, Jul 05 2006
G.f.: (Product_{k>0} (1 - x^(4*k))^2 / (1 - x^(2*k - 1)))^2.
a(n) = A000203(A004767(n))/4. - Michel Marcus, Nov 30 2015
Sum_{k=0..n} a(k) = (Pi^2/16) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 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

A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 06 2007

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

			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 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033686, A098098, A134078, A227696, A263773.

Cf. A000700, A000122, A004016, A005882, A005928, A010054, A121373.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* Michael Somos, Feb 19 2015 *)

{a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^3 )^2, n))};

{a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* Michael Somos, Feb 19 2015 */

Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.
Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.
a(n) = (-1)^n * A033686(n). 18 * a(n) = A134078(3*n + 2).
From Michael Somos, Feb 19 2015: (Start)
a(2*n + 1) = -2 * A098098(n).
Convolution square of A227696. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - Amiram Eldar, Nov 23 2023

A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 3, 0, -1, 0, 7, -8, 6, 0, 1, 0, 8, -12, 15, 0, -7, 0, 18, -16, 12, 0, 5, 0, 14, -26, 24, 0, -6, 0, 31, -24, 18, 0, -5, 0, 20, -28, 42, 0, -8, 0, 36, -48, 24, 0, 13, 0, 31, -36, 42, 0, -25, 0, 56, -40, 30, 0, 6, 0, 32, -64, 63, 0, -12, 0, 54, -48, 48, 0

Offset: 2

Michael Somos, Sep 26 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q^2 - 2*q^3 + 3*q^4 - q^6 + 7*q^8 - 8*q^9 + 6*q^10 + q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 2..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008438, A033686, A093829, A144614, A229616.

Basis( ModularForms( Gamma0(6), 2), 70)[3] /* Michael Somos, Mar 05 2023 */

a[ n_] := If[n < 1, 0, Sum[ {0, 1, -2, 1, 0, 0}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, 0, Sum[ {0, 1/2, -2/3, 1/2, 0, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)]^6 / EllipticTheta[ 2, 0, q^(1/2)]^2 / 16, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv( n, d, n/d * [0, 0, 1, -2, 1, 0][d%6 + 1]))};

{a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 1/2, -2/3, 1/2, 0][d%6 + 1]))};

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3))^2, n))};

ModularForms( Gamma0(6), 2, prec=70).2;

Expansion of (a(q) - a(q^2))^2 / 36 = c(q^2)^4 / (9 * c(q)^2) in powers of q where a(), c() are cubic AGM theta functions.
Expansion of ((eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3))^2 in powers of q.
Euler transform of period 6 sequence [ -2, 2, 4, 2, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/12) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229616.
G.f.: sum_{k>0} x^(6*k-4) / (1 - x^(6*k-4))^2 - 2 * x^(6*k-3) / (1 - x^(6*k-3))^2 + x^(6*k-2) / (1 - x^(6*k-2))^2.
G.f.: sum_{k>0} (3*k-2) * x^(6*k-4) / (1 - x^(6*k-4)) - (4*k-2) * x^(6*k-3) / (1 - x^(6*k-3)) + (3*k-1) * x^(6*k-2) / (1 - x^(6*k-2)).
a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 2) = A144614(n). a(6*n + 3) = -2 * A008438(n). a(6*n + 4) = 3 * A033686(n).
Convolution square of A093829.

A281722 Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

Original entry on oeis.org

1, 3, -18, 12, 21, -36, 36, 24, -90, 12, 54, -72, 84, 42, -144, 72, 93, -108, 36, 60, -252, 96, 108, -144, 180, 93, -252, 12, 168, -180, 216, 96, -378, 144, 162, -288, 84, 114, -360, 168, 270, -252, 288, 132, -504, 72, 216, -288, 372, 171, -558, 216, 294, -324

Offset: 0

Michael Somos, Jan 28 2017

sign

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

			G.f. = 1 + 3*q - 18*q^2 + 12*q^3 + 21*q^4 - 36*q^5 + 36*q^6 + 24*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from G. C. Greubel)

Cf. A004016, A005928, A033686, A096726, A144614.

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

{a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A)^3 * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A)^2, n))};

Convolution of the sequences A004016 and A005928.
The g.f. is the product of the g.f.'s for A004016 and A005928. - N. J. A. Sloane, Jan 30 2017
Expansion of eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144614.
a(3*n + 2) = A096726(3*n + 2) - 27 * A033686(n). a(n) == A096726(n) (mod 27). - Michael Somos, Sep 04 2017

A144615 a(n) = A000203(3n+2).

Original entry on oeis.org

3, 6, 15, 12, 24, 18, 42, 24, 42, 30, 63, 48, 60, 42, 84, 48, 93, 54, 120, 60, 96, 84, 126, 72, 114, 96, 186, 84, 132, 90, 168, 120, 171, 102, 210, 108, 216, 114, 210, 144, 186, 156, 255, 132, 204, 138, 336, 168, 222, 150, 300, 192, 240, 192, 294, 168, 324, 174, 372, 180, 336

Offset: 0

N. J. A. Sloane, Jan 15 2009

nonn

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

			G.f. = 3 + 6*x + 15*x^2 + 12*x^3 + 24*x^4 + 18*x^5 + 42*x^6 + 24*x^7 + 42*x^8 + ...
G.f. = 3*q^2 + 6*q^5 + 15*q^8 + 12*q^11 + 24*q^14 + 18*q^17 + 42*q^20 + 24*q^23 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..10000

Cf. A000203, A016789, A033686.

sequence := List([0..10^4],n->Sigma(3*n+2)); # Muniru A Asiru, Dec 29 2017

[SumOfDivisors(3*n+2): n in [0..70]]; // Vincenzo Librandi, Jan 19 2018

with(numtheory):
seq(sigma(3*n+2), n=0..10^3); # Muniru A Asiru, Dec 29 2017

a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 3 n + 2]]; (* Michael Somos, Jul 14 2015 *)
a[ n_] := SeriesCoefficient[ 3 (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Jul 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Jun 07 2012 */

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

Expansion of q^(-2/3) * c(q)^2 / 3 in powers of q where c() is a cubic AGM theta function. - Michael Somos, Jun 07 2012
Expansion of q^(-2/3) * 3 * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Jun 07 2012
a(n) = A000203(A016789(n)). - Michel Marcus, Jul 14 2015
a(n) = 3*A033686(n). - Robert G. Wilson v, Jan 12 2018
Sum_{k=1..n} a(k) = (2*Pi^2/9) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A232343 Expansion of q^(-5/3) * c(q^2)^3 / (9 * c(q)) in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 2, 0, 3, -2, 4, 0, 5, -5, 8, 0, 7, -4, 8, 0, 9, -8, 10, 0, 14, -6, 12, 0, 16, -14, 14, 0, 15, -8, 20, 0, 17, -14, 18, 0, 19, -10, 24, 0, 26, -21, 22, 0, 23, -16, 28, 0, 25, -20, 32, 0, 32, -14, 28, 0, 29, -28, 30, 0, 38, -16, 32, 0, 33, -31, 40, 0, 40

Offset: 0

Michael Somos, Nov 22 2013

sign

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

			G.f. = 1 - x + 2*x^2 + 3*x^4 - 2*x^5 + 4*x^6 + 5*x^8 - 5*x^9 + 8*x^10 + ...
G.f. = q^5 - q^8 + 2*q^11 + 3*q^17 - 2*q^20 + 4*q^23 + 5*q^29 - 5*q^32 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A033686, A098098.

Basis( ModularForms( Gamma0(18), 2), 210) [6]; /* Michael Somos, Jul 09 2018 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^9 / (QPochhammer[ x^2] QPochhammer[ x^3])^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] QPochhammer[ x^12])^3 / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# == 2, 2 - 2^#2,# == 3, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[3 n + 5]) / 6]; (* Michael Somos, Jul 09 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A))^3, n))};

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 5; A = factor(n); 1/6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 2 - 2^e, p==3, 0, (p^(e+1) - 1) / (p - 1))))};

Expansion of q^(-5/3) * eta(q) * eta(q^6)^9 / (eta(q^2) * eta(q^3))^3 in powers of q.
Euler transform of period 6 sequence [-1, 2, 2, 2, -1, -4, ...].
a(n) = 1/6 * b(3*n + 5) where b() is multiplicative with b(2^e) = 2 - 2^e, b(3^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
a(2*n) = A098098(n). a(4*n + 1) = - A033686(n). a(4*n + 3) = 0.

A242874 Expansion of b(q)^2 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 9, 12, -42, 18, 36, -48, 45, 12, -108, 36, 84, -84, 72, 72, -186, 54, 36, -120, 126, 96, -216, 72, 180, -186, 126, 12, -336, 90, 216, -192, 189, 144, -324, 144, 84, -228, 180, 168, -540, 126, 288, -264, 252, 72, -432, 144, 372, -342, 279, 216, -588

Offset: 0

Michael Somos, May 26 2014

sign

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

			G.f. = 1 - 6*q + 9*q^2 + 12*q^3 - 42*q^4 + 18*q^5 + 36*q^6 - 48*q^7 + 45*q^8 + ...

O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See t, page 1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

Cf. A005928, A008653, A033686, A144614.

A := Basis( ModularForms( Gamma0(9), 2), 53); A[1] - 6*A[2] + 9*A[3]; /* Michael Somos, Sep 27 2016 */

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

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

A = ModularForms( Gamma0(9), 2, prec=53) . basis(); A[0] - 6*A[1] + 9*A[2];

Expansion of (eta(q)^3 / eta(q^3))^2 in powers of q.
Euler transform of period 3 sequence [-6, -6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 243 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033686.
G.f.: Product_{k>0} ( (1 - x^k)^3 / (1 - x^(3*k)) )^2.
a(3*n) = A008653(n). a(3*n + 1) = -6 * A144614(n). a(3*n + 2) = 9 * A033686(n).
Convolution square of A005928.

A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31

Offset: 0

Michael Somos, Nov 12 2018

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 124 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = x^3 - x^4 + 2*x^6 - 3*x^8 + 4*x^9 + x^12 + 6*x^15 - 7*x^16 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..100000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A008438, A033686, A097723, A112610, A144614, A224226, A229615, A321528.

A := Basis( ModularForms( Gamma0(12), 2), 76); A[4] - A[5];

a[ n_] := SeriesCoefficient[ x^3 QPochhammer[ x, x^2] QPochhammer[ x^12]^6 / (QPochhammer[ x^3, x^6]^3 QPochhammer[ x^4]^2), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, 0, s[n/3] - s[n/4] - s[n/6] + s[n/12]]];
a[ n_] := If[ n < 1, 0, Sum[ d {0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0, 0}[[Mod[d, 12, 1]]] / 12, {d, Divisors[n]}]];

{a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0][d%12 + 1] / 12))};

{a(n) = my(s = x -> if( frac(x), 0, sigma(x))); if( n<1, 0, s(n/3) - s(n/4) - s(n/6) + s(n/12))};

{a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^12 + A)^6 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};

Expansion of x^3 * (psi(x^3) * psi(x^6))^3 / (psi(x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
Expansion of x^3 * chi(-x) * f(-x^12)^6 / (chi(-x^3)^3 * f(-x^4)^2) in powers of x where chi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321528.
a(n) = s(n/3) - s(n/4) - s(n/6) + s(n/12) where s(x) = sum of divisors of x for integer x else 0.
a(6*n + 1) = a(6*n + 5) = a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A224226(n) if n>0. a(2*n) = -A229615(n). a(6*n + 3) = A008438(n). a(12*n + 6) = 2*A008438(n).
a(12*n + 3) = A112610(n). a(12*n + 4) = -A144614(n). a(12*n + 8) = -3*A033686(n). a(12*n + 9) = 4*A097723(n).

cp's OEIS Frontend

A098098 a(n) = sigma(6*n+5)/6.

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A097723 One fourth of sum of divisors of 4n+3.

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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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A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

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A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.

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A281722 Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

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