A123330 -id:A123330 - cp's OEIS frontend

A107760 Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.

1, 3, 3, 3, 3, 0, 3, 6, 3, 3, 0, 0, 3, 6, 6, 0, 3, 0, 3, 6, 0, 6, 0, 0, 3, 3, 6, 3, 6, 0, 0, 6, 3, 0, 0, 0, 3, 6, 6, 6, 0, 0, 6, 6, 0, 0, 0, 0, 3, 9, 3, 0, 6, 0, 3, 0, 6, 6, 0, 0, 0, 6, 6, 6, 3, 0, 0, 6, 0, 0, 0, 0, 3, 6, 6, 3, 6, 0, 6, 6, 0, 3, 0, 0, 6, 0, 6, 0, 0, 0, 0, 12, 0, 6, 0, 0, 3, 6, 9, 0, 3, 0, 0, 6, 6

Offset: 0

Michael Somos, May 24 2005

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 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.42).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A097195, A112604, A112605, A123330, A132973, A132979.
Cf. A112606, A112607, A112608, A112609. - Michael Somos, Sep 27 2013
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma1(6), 1), 88); A[1] + 3*A[2]; /* Michael Somos, Aug 04 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 3 Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* Michael Somos, Aug 04 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^3 / (4 EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
QP = QPochhammer; s = QP[q^3]*(QP[q^2]^6/(QP[q]^3*QP[q^6]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<1, n==0, 3 * direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X)))[n])};

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

{a(n) = if ( n<1, n==0, 3 * sumdiv( n, d, kronecker( -12, d)))};

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 3*A[1] # Michael Somos, Sep 27 2013

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u^2*w + 4 * v*w^2 - 4 * v^2*w - 2 * u*v*w.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u2) * (u1 - u2 - u3 + u6) - 3 * u6 * (u2 - u6).
Expansion of psi(q)^3 / psi(q^3) in powers of q where psi() is a Ramanujan theta function.
Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Aug 30 2008
Euler transform of period 6 sequence [ 3, -3, 2, -3, 3, -2, ...].
Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - Michael Somos, Aug 11 2009
a(n) = 3 * b(n) unless n=0 and b() is multiplicative with b(p^e) = 1 if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6). - Michael Somos, Aug 11 2009
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123330. - Michael Somos, Aug 11 2009
G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1)))^3 / (Product_{k>0} (1 - x^(6*k)) / (1 - x^(6*k - 3))). - Michael Somos, Aug 11 2009
a(n) = 3 * A035178(n) unless n=0. a(n) = (-1)^n * A132973. a(2*n) = a(3*n) = a(n). a(6*n + 5) = 0. a(2*n + 1) = 3 * A033762. a(3*n + 1) = 3 * A033687(n). a(4*n + 1) = 3 * A112604(n). a(4*n + 3) = 3 * A112605(n). a(6*n + 1) = 3 * A097195(n). Convolution inverse of A132979.
a(8*n + 1) = 3 * A112606(n). a(8*n + 3) = 3* A112608(n). a(8*n + 5) = 6 * A112607(n-1). a(8*n + 7) = 6 * A112609(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3)/2 = 2.720699... . - Amiram Eldar, Dec 28 2023

A123633 Expansion of (c(q^2)/c(q))^3 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 3, 5, -18, 15, 24, -75, 57, 86, -252, 183, 262, -744, 522, 725, -1998, 1365, 1852, -4986, 3336, 4436, -11736, 7719, 10103, -26322, 17067, 22040, -56682, 36306, 46336, -117867, 74700, 94378, -237744, 149277, 186926, -466836, 290706, 361126, -895014, 553224

Offset: 1

Michael Somos, Oct 03 2006, Jan 21 2009

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).
In the arXiv:2305.13951 paper on page 21 is this: "The q-expansion of y coincides with the sequence A123633 in the OEIS". - Michael Somos, May 26 2023

			G.f. = q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + 57*q^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Xuhang Jiang, Xing Wang, Li Lin Yang and Jing-Bang Zhao, Epsilon-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves, arXiv:2305.13951 [hep-th], 2023. See page 21.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A092848, A093829, A105559, A123330, A128636, A128642.

M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A := M2/(M1 + 2*M2); A; /* Michael Somos, May 26 2023 */

a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 1, n, 2}] / Product[ 1 - q^k, {k, 3, n, 6}]^3)^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3 * (eta(x^6 + A) / eta(x^3 + A))^9, n))};

Expansion of q / (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ -3, 0, 6, 0, -3, 0, ...].
G.f. A(x) satisfies  0 = f(A(x), A(x^2)) where f(u, v)=  u^2 - v - u*v * (6 + 8*v).
G.f.: x * (Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3 )^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128642.
A128636(n) = a(n) unless n = 0. Convolution inverse of A105559.
Convolution cube of A092848.
Convolution with A123330 is A093829. - Michael Somos, May 26 2023

A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

Original entry on oeis.org

1, 2, 1, 1, 0, 2, 2, 2, 1, 0, 0, 1, 2, 4, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 1, 4, 1, 2, 0, 0, 2, 2, 0, 0, 0, 1, 2, 4, 2, 0, 0, 4, 2, 0, 0, 0, 0, 1, 3, 2, 0, 2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 1, 2, 0, 4, 2, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 2, 6, 0, 1, 0, 0, 2, 4, 0

Offset: 1

Michael Somos, Sep 26 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A123330(n)=2*a(n) if n>0. A113974(n)=-(-1)^n*a(n).

Cf. A248897.

f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (3-(-1)^e)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

{a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*kronecker(-3,d)))}

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

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)^6/ eta(x+A)^2/eta(x^6+A)^3-1)/2, n))}

Moebius transform is period 6 sequence [ 1, 1, 0, -1, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (3-(-1)^e)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3n) = a(4n) = a(n). a(6n+5) = 0.
G.f.: Sum_{k>0} x^k/(1-x^k+x^(2k)) = (theta_3(-q^3)^3/theta_3(-q) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Nov 14 2023

A132000 Expansion of (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

Original entry on oeis.org

1, -1, -5, -1, 11, 24, -5, -50, -53, -1, 120, 120, 11, -170, -250, 24, 203, 288, -5, -362, -264, -50, 600, 528, -53, -601, -850, -1, 550, 840, 120, -962, -821, 120, 1440, 1200, 11, -1370, -1810, -170, 1272, 1680, -250, -1850, -1320, 24, 2640, 2208, 203, -2451

Offset: 0

Michael Somos, Aug 06 2007

sign

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

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

			G.f. = 1 - x - 5*x^2 - x^3 + 11*x^4 + 24*x^5 - 5*x^6 - 50*x^7 - 53*x^8 - x^9 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).

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

Cf. A113261, A122373, A123330, A131943.

A := Basis( ModularForms( Gamma1(6), 3), 50); A[1] - A[2] - 5*A[3] - A[4]; /* Michael Somos, Nov 03 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[(9 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^5 - EllipticTheta[ 4, 0, q]^5 EllipticTheta[ 4, 0, q^3]) / 8, {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, d^2 * (-1)^d * kronecker(-3, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A)^5 / eta(x^6 + A)^4, n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, 2 + ((-4)^(e+1) - 1) / 5, p = p^2 * kronecker(-3, p); (p^(e+1) - 1) / (p-1) )))};

Expansion of phi(-q)^2 * phi(-q^3)^2 * psi(q)^3 / psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
Euler transform of period 6 sequence [-1, -5, -6, -5, -1, -6, ...].
a(n) = -b(n) where b() is multiplicative with b(2^e) = 2+((-4)^(e+1)-1)/5, b(3^e) = 1, b(p^e) = (q^(e+1) - 1) / (q-1) where q = p^2*Kronecker(-3, p) if p > 3.
a(3*n) = a(n).
G.f.: 1 - Sum_{k>0} k^2 * Kronecker(-3, k) * x^k / (1 - (-x)^k) = Product_{k>0} (1  - x^(3*k)) * (1 - x^k)^5 / (1 - x^k + x^(2*k))^4.
a(n) = (-1)^n * A113261(n). Convolution of A123330 and A131943.
a(n) = -A132000(n) unless n=0.
Expansion of (9 * phi(-q) * phi(-q^3)^5 - phi(-q)^5 * phi(-q^3)) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 03 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 15552^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122373. - Michael Somos, Nov 03 2015

A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 12, 20, 28, 24, 28, 32, 60, 68, 72, 48, 44, 56, 96, 120, 124, 72, 76, 80, 168, 160, 144, 96, 76, 124, 168, 212, 224, 120, 168, 128, 252, 240, 216, 192, 92, 152, 240, 280, 360, 168, 224, 176, 336, 408, 288, 192, 140, 228, 372, 360, 392, 216, 220, 288

Offset: 0

Michael Somos, Sep 19 2013

nonn

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 13 and 34 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

			G.f. = 1 + 4*q + 12*q^2 + 20*q^3 + 28*q^4 + 24*q^5 + 28*q^6 + 32*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
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. A123330, A227229.

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 4*A[2] + 12*A[3];

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3]^6 / (QPochhammer[ q]^2 QPochhammer[ q^6]^3))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^6 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 4/3, 1, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 2, 1, 1, -6}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];

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

A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 4*A[1] + 12*A[2];

Expansion of (a(q) + 2*a(q^2))^2 / 9 in powers of q where a(q) is a cubic AGM theta function.
Expansion of c(q)^4 / (3 * c(q^2))^2 in powers of q where c(q) is a cubic AGM theta function.
Expansion of (eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3))^2 in powers of q.
Euler transform of period 6 sequence [4, 2, -8, 2, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227229.
Convolution square of A123330.

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, 2, -4, 0, 4, -8, 4, 2, 0, 0, 2, -8, 8, 0, -4, 0, 4, -8, 0, 4, 0, 0, 4, -4, 8, 2, -8, 0, 0, -8, 4, 0, 0, 0, 2, -8, 8, 4, 0, 0, 8, -8, 0, 0, 0, 0, 2, -12, 4, 0, -8, 0, 4, 0, 8, 4, 0, 0, 0, -8, 8, 4, -4, 0, 0, -8, 0, 0, 0, 0, 4, -8, 8, 2, -8, 0, 8, -8, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -16, 0, 4, 0, 0, 4, -8

Offset: 0

Michael Somos, Jul 29 2011

sign

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

			G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033687, A033762, A097195, A112604, A112605.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

{a(n) = if( n<1, n==0, 2 * if( n%3==1, -2, 1) * sumdiv( n, d, -(-1)^d * kronecker( -3, d)))};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)), n))};

Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).

cp's OEIS Frontend

A107760 Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.

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A123633 Expansion of (c(q^2)/c(q))^3 in powers of q where c() is a cubic AGM theta function.

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A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

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A132000 Expansion of (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

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A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 where phi() is a Ramanujan theta function.

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A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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