A109041 -id:A109041 - cp's OEIS frontend

A205973 a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

1, -9, 27, -18, -351, 1080, 216, -5850, 9639, -306, -35640, 96120, -16848, -356490, 508950, 131760, -1821015, 4139424, 69768, -13621698, 18996120, -4925700, -57383640, 136178064, 21282912, -405810225, 557193870, -1767762, -1859194350, 3887571240, -539161920

Offset: 0

Paul D. Hanna, Feb 04 2012

sign

Compare the g.f. to the Lambert series of A109041:

1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).

			G.f.: A(x) = 1 - 9*x + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
where A(x) = 1 - 1*9*x + 1*27*x^2 - 2*9*x^3 - 3*117*x^4 + 5*216*x^5 + 8*27*x^6 - 13*450*x^7 + 21*459*x^8 +...+ Fibonacci(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-x-x^2) - 1*4*x^2/(1-3*x^2+x^4) + 3*16*x^4/(1-7*x^4+x^8) - 5*25*x^5/(1-11*x^5-x^10) + 13*49*x^7/(1-29*x^7-x^14) - 21*64*x^8/(1-47*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).

Cf. A209453 (Pell variant).

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 9*sum(m=1,n,fibonacci(m)*kronecker(m,3)*m^2*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

Original entry on oeis.org

1, -9, 54, -45, -1404, 6264, 1890, -76050, 187272, -8865, -1540944, 6200280, -1621620, -51195330, 109055700, 42125400, -868685040, 2946297888, 74093670, -21584605122, 44912353824, -17376284250, -302040439920, 1069478852112, 249392931480, -7095191496489

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare the g.f. to the Lambert series of A109041:

1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).

			G.f.: A(x) = 1 - 9*x + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.

A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 9*sum(m=1,n,Pell(m)*kronecker(m,3)*m^2*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

Original entry on oeis.org

1, 3, 9, 13, 24, 27, 50, 51, 81, 72, 120, 117, 170, 150, 216, 205, 288, 243, 362, 312, 450, 360, 528, 459, 601, 510, 729, 650, 840, 648, 962, 819, 1080, 864, 1200, 1053, 1370, 1086, 1530, 1224, 1680, 1350, 1850, 1560, 1944, 1584, 2208, 1845, 2451, 1803, 2592

Offset: 1

Michael Somos, May 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 3 of the 74 eta-quotients listed in Table I of Martin (1996).
a(n+1) is the number of partition triples of n where each partition is 3-core (see Theorem 3.1 of Wang link).
Convolution cube of A033687.
Convolution square is A198958. - Michael Somos, Dec 26 2015

			G.f. = q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 + ...

George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001). See p. 314, Eq. (14.2.14).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vaclav Kotesovec)
Jonathan M. Borwein and Peter 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). See page 697.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A004016, A005882, A005928, A033687, A109041, A129404, A198958, A231947.

A := Basis( ModularForms( Gamma1(3), 3), 52); A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, #^2 KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3]^3 / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^9 / (1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

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

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

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

a(n) = sumdiv(n, d, ((d % 3) == 1)*(n/d)^2) - sumdiv(n, d, ((d % 3)== 2)*(n/d)^2); \\ Michel Marcus, Jul 14 2015

Expansion of (c(q) / 3)^3 in powers of q where c(q) is a cubic AGM theta function.
Euler transform of period 3 sequence [ 3, 3, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 6*u*v*w + 8*u*w^2 - u^2*w.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) = x * Product_{k>0} (1 - x^(3*k))^9 / (1 - x^k)^3.
a(n) is multiplicative and a(p^e) = ((p^2)^(e+1) - u^(e+1)) / (p^2 - u) where u = 0, 1, -1 when p == 0, 1, 2 (mod 3). - Michael Somos, Oct 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109041.
a(3*n) = 9 * a(n). a(3*n + 1) = A231947(n). -  Michael Somos, May 18 2015
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 4*Pi^3/(81*sqrt(3)) = 0.8840238... (A129404). - Amiram Eldar, Nov 09 2023

A231947 Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 13, 50, 72, 170, 205, 362, 360, 601, 650, 962, 864, 1370, 1224, 1850, 1584, 2451, 2210, 2880, 2520, 3722, 3277, 4490, 3600, 5330, 4706, 6242, 5040, 6912, 6120, 8500, 6624, 9410, 7813, 10610, 8424, 11882, 10250, 12672, 10440, 14521, 12506, 16130, 12240

Offset: 0

Michael Somos, Nov 15 2013

nonn

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

			G.f. = 1 + 13*x + 50*x^2 + 72*x^3 + 170*x^4 + 205*x^5 + 362*x^6 + 360*x^7 + ...
G.f. = q + 13*q^4 + 50*q^7 + 72*q^10 + 170*q^13 + 205*q^16 + 362*q^19 + ...

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

Cf. A109041, A231948.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)*(eta[q]^3 + 9*eta[q^9]^3)^2*eta[q^3]/eta[q], {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

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

Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3)^2 * eta(q^3) / eta(q) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231948.
-9 * a(n) = A109041(3*n + 1).

A103440 a(n) = Sum[d|n, d==1 (mod 3), d^2] - Sum[d|n, d==2 (mod 3), d^2].

Original entry on oeis.org

1, -3, 1, 13, -24, -3, 50, -51, 1, 72, -120, 13, 170, -150, -24, 205, -288, -3, 362, -312, 50, 360, -528, -51, 601, -510, 1, 650, -840, 72, 962, -819, -120, 864, -1200, 13, 1370, -1086, 170, 1224, -1680, -150, 1850, -1560, -24, 1584, -2208, 205, 2451, -1803, -288, 2210, -2808, -3, 2880, -2550

Offset: 1

Ralf Stephan, Feb 11 2005

			G.f. = q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 - 51*q^8 + q^9 + ...

Robert Israel, Table of n, a(n) for n = 1..10000
G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Equals A103637(n) - A103638(n). Cf. A002173.

A109041(n) = -9 * a(n) unless n=0. A014985(n) = a(2^n). -24 * A134340(n) = a(6*n+5).

f:= proc(n) local D,d;
  D:= numtheory:-divisors(n/3^padic:-ordp(n,3));
  -add((-1)^(d mod 3)*d^2, d = D)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 16 2018

a[n_] := Sum[m=Mod[d, 3]; (Boole[m==1]-Boole[m==2]) d^2, {d, Divisors[n]}];
Array[a, 56] (* Jean-François Alcover, Aug 16 2018 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ x]^9 / QPochhammer[ x^3]^3) / 9, {x, 0, n}]; (* Michael Somos, Sep 07 2018 *)

{a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -3, d)))}; /* Michael Somos, Oct 21 2007 */

{a(n) = my(A, p, e, a0, a1, x, y, z); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, z = kronecker( -3, p) * p^2; a0 = 1; a1 = y = z + 1; for(i=2, e, x = y * a1 - z * a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Oct 21 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^9 / eta(x^3 + A)^3) / 9, n))}; /* Michael Somos, Oct 21 2007 */

G.f.: F(q) = Sum_{n>=1} A049347(n-1) * n^2 * q^n / (1 - q^n).
G.f.: F(q) = -q * G'(q) / (9*G(q)), with G(q) = Product_{n>=1} (1 - q^n)^(9*n * A049347(n-1)).
a(n) is multiplicative with a(3^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - z * a(p^(e-2)) where z = Kronecker(-3, p) * p^2 and a(p) = z + 1.
a(3*n) = a(n).
G.f.: Sum_{k>0} x^k * (1 - x^k - 6*x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3. - Michael Somos, Oct 21 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v + w + 3*v^2 - 8*w^2 + 6*v*w - 8*u*w + 6*u*v - 9*v^3 - 54*u*v*w + 72*u*w^2 - 9*u^2*w. - Michael Somos, Dec 23 2007

A134340 Expansion of psi(x)^3 * f(-x^3)^3 / chi(-x)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 5, 12, 22, 35, 50, 70, 92, 117, 145, 170, 210, 250, 287, 330, 362, 425, 477, 532, 600, 626, 715, 782, 850, 925, 962, 1100, 1162, 1247, 1335, 1370, 1520, 1617, 1750, 1810, 1850, 2040, 2147, 2262, 2380, 2451, 2625, 2752, 2882, 3015, 3005, 3290, 3500, 3577

Offset: 0

Michael Somos, Oct 21 2007

nonn

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

			G.f. = 1 + 5*x + 12*x^2 + 22*x^3 + 35*x^4 + 50*x^5 + 70*x^6 + 92*x^7 + 117*x^8 + ...
G.f. = q^5 + 5*q^11 + 12*q^17 + 22*q^23 + 35*q^29 + 50*q^35 + 70*q^41 + 94*q^47 + ...

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. A103440, A109041.

a[ n_] := If[ n < 0, 0, (-1/24) DivisorSum[ 6 n + 5, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Oct 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x^(1/2)]^3 / (8 x^(3/8)), {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

{a(n) = if( n<0, 0, n = 6*n + 5; sumdiv(n, d, d^2 * kronecker( -3, d)) / -24 )};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^3 / eta(x + A)^5, n))};

Expansion of q^(-5/6) * eta(q^2)^8 * eta(q^3)^3 / eta(q)^5 in powers of q.
Euler transform of period 6 sequence [ 5, -3, 2, -3, 5, -6, ...].
-24 * a(n) = A103440(6*n + 5). 216 * a(n) = A109041(6*n + 5).

A231948 Expansion of a(q)^2 * b(q) in powers of q where a(), b() are cubic AGM theta functions.

Original entry on oeis.org

1, 9, 0, -90, 117, 0, -216, 450, 0, -738, 648, 0, -1170, 1530, 0, -1728, 1845, 0, -2160, 3258, 0, -4500, 3240, 0, -3672, 5409, 0, -6570, 5850, 0, -6480, 8658, 0, -8640, 7776, 0, -9594, 12330, 0, -15300, 11016, 0, -10800, 16650, 0, -17280, 14256, 0, -18450

Offset: 0

Michael Somos, Nov 15 2013

sign

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

			G.f. = 1 + 9*q - 90*q^3 + 117*q^4 - 216*q^6 + 450*q^7 - 738*q^9 + ...

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

Cf. A109041, A181976, A231947.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3 + 9*eta[q^9]^3)^2*(eta[q]/eta[q^3])^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

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

Expansion of (eta(q)^3 + 9 * eta(q^9)^3)^2 * (eta(q) / eta(q^3))^3 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(11/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231947.
a(3*n + 2) = 0.  a(3*n + 1) = 9 * A231947(n). 3 * A109041(n) = a(3*n) + A109041(3*n) + A181976(3*n).

A215711 Expansion of a(q) * b(q)^3 in powers of q where a(), b() are cubic AGM theta functions.

Original entry on oeis.org

1, -3, -27, 159, -219, -378, 1431, -1032, -1755, 4533, -3402, -3996, 11607, -6594, -9288, 20034, -14043, -14742, 40797, -20580, -27594, 54696, -35964, -36504, 93015, -47253, -59346, 122631, -75336, -73170, 180306, -89376, -112347, 211788, -132678, -130032

Offset: 0

Michael Somos, Aug 21 2012

sign

			G.f. = 1 - 3*q - 27*q^2 + 159*q^3 - 219*q^4 - 378*q^5 + 1431*q^6 - 1032*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010. See Table 1, p.12

Cf. A004016, A109041, A133078, A198956, A215690.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(1 + 9*(eta[q^9]/eta[q])^3)*(eta[q]^3/eta[q^3])^4, {q, 0, 50}], q] (* G. C. Greubel, Aug 10 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3) * (eta(x + A)^3 / eta(x^3 + A))^4, n))}

Expansion of (1 + 9 * q * (eta(q^9) / eta(q))^3) * (eta(q)^3 / eta(q^3))^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A198956.
G.f.: 1 - 3 * (Sum_{k>0} k^3 * x^k / (1 - x^k) - 3 * (3*k)^3 * x^(3*k) / (1 - x^(3*k))).
Convolution of A215690 and A133078. Convolution of A004016 and A109041.

A320677 Expansion of s(q)^6 where s() is cubic AGM theta functions.

Original entry on oeis.org

1, -18, 135, -504, 657, 2052, -10071, 12384, 20277, -83610, 72090, 122040, -355581, 245124, 379512, -1050624, 770589, 966492, -2700081, 1724616, 2287062, -5636880, 3616164, 4471632, -11385657, 6820722, 8554194, -19963440, 12302568, 14113332, -34631226, 19737936

Offset: 0

Seiichi Manyama, Oct 19 2018

sign

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

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

s(q)^m: A005928 (m=1), A242874 (m=2), A109041 (m=3), A133078 (m=4), this sequence (m=6).

Cf. A004010, A109041.

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

cp's OEIS Frontend

A205973 a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

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A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

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A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

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A231947 Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.

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A103440 a(n) = Sum[d|n, d==1 (mod 3), d^2] - Sum[d|n, d==2 (mod 3), d^2].

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A134340 Expansion of psi(x)^3 * f(-x^3)^3 / chi(-x)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions.

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A231948 Expansion of a(q)^2 * b(q) in powers of q where a(), b() are cubic AGM theta functions.