A033715 -id:A033715 - cp's OEIS frontend

A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

1, 6, 18, 44, 90, 144, 212, 288, 330, 418, 528, 588, 836, 1008, 1056, 1440, 1386, 1356, 1894, 1644, 2064, 2880, 2484, 3168, 3428, 2838, 3696, 3864, 4128, 5040, 5280, 5760, 5418, 5656, 5988, 5376, 7678, 8208, 7572, 10080, 8208, 7788, 10560, 8652, 10404, 13104

Offset: 0

Michael Somos, Dec 10 2007

nonn

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

			G.f. = 1 + 6*q + 18*q^2 + 44*q^3 + 90*q^4 + 144*q^5 + 212*q^6 + 288*q^7 + ...

Vaclav Kotesovec, 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. A029713, A030207, A033715, A097057.

A := Basis( ModularForms( Gamma1(8), 3), 46);  A[1] + 6*A[2] + 18*A[3] + 44*A[4] + 90*A[5] + 144*A[6] + 212*A[7]; /* Michael Somos, Oct 14 2015 */

nmax=60; CoefficientList[Series[Product[((1-x^k)^2 * (1+x^k)^4 * (1+x^(2*k)) / (1+x^(4*k))^2)^3,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^3, {q, 0, n}]; (* Michael Somos, Oct 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^4 + A))^3 / (eta(x + A) * eta(x^8 + A))^2)^3, n))};

Expansion of (eta(q^2) * eta(q^4))^9 / (eta(q) * eta(q^8))^6 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1/(8 t)) = 2^(9/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - x^k)^2 * (1 + x^k)^4 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^3.
a(n) = A029713(n) + 6 * A030207(n). Convolution of A033715 and A097057.
a(n) = A028578(4*n). - Michael Somos, Oct 14 2015

A226240 Expansion of phi(q^4) * phi(q^8) + 2 * q phi(q^2) psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 0, 4, 0, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 0, 0, 6, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 0, 8, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 0, 4, 0, 0

Offset: 0

Michael Somos, Jun 01 2013

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

			G.f. = 1 + 2*q + 4*q^3 + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + 2*q^16 + ...

Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033715, A113411.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(32), 1), 87); A[1] + 2*A[2] + 4*A[4] + 2*A[5] + 2*A[9] + 6*A[10] + 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Jun 18 2014 */

a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ Boole[ 2 != Mod[ k, 4]] q^k (1 + q^(2 k)) / (1 + q^(4 k)), {k, n}], {q, 0, n}];

{a(n) = if( n<1, n==0, 2 * (n%4 != 2) * sumdiv( n, d, kronecker( -2, d)))};

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, e>1, if( p%8<5, e+1, (1 + (-1)^e) / 2)))))};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^16 + A))^3 / (eta(x^4 + A) * eta(x^32 + A))^2 + 2 * x * eta(x^4 + A)^5 * eta(x^16 + A)^2 / (eta(x^2 + A)^2 * eta(x^8 + A)^3), n))};

Expansion of phi(q) * phi(q^8) + 4 * q^3 * psi(q^8) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2) = 0, b(2^e) = 1 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
G.f.: 1 + 2 * Sum_{k>0 & k !=2 (mod 4)} q^k * (1 + q^(2*k)) / (1 + q^(4*k)).
a(4*n + 2) = 0. a(2*n + 1) = 2 * A113411(n). a(4*n) = A033715(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/(4*sqrt(2)) = 1.6660811... . - Amiram Eldar, Dec 29 2023

A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, 1, 4, -2, 0, 1, -1, 4, -2, -2, 4, 0, 0, 1, 2, -1, -2, 4, 0, -2, 0, -2, 5, 4, -4, 0, 4, 0, 0, 1, -4, 2, 0, -1, 4, -2, 0, 4, 2, 0, -2, -2, 4, 0, 0, -2, 1, 5, -4, 4, 4, -4, 0, 0, -4, 4, -2, 0, 4, 0, 0, 1, 8, -4, -2, 2, 0, 0, 0, -1, 2, 4, -2, -2, 0, 0

Offset: 1

Michael Somos, Jun 30 2014

sign

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

			G.f. = q + q^2 - 2*q^3 + q^4 + 4*q^5 - 2*q^6 + q^8 - q^9 + 4*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A107635, A121373, A143259, A243747, A244560.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];

a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * (A - subst(A, x, x^2)) / 2, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];

Expansion of q * f(-q, -q^7)^2 * phi(q) / psi(-q) = q * f(-q, -q^7)^2 * chi(q)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].
Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].
Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.
a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).
a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 - 1/sqrt(2))/2 = 0.460075... . - Amiram Eldar, Jun 08 2025

A245572 Expansion of phi(q) * phi(q^2) + 2 * phi(-q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

3, 2, -2, 4, 6, 0, -4, 0, 6, 6, 0, 4, 12, 0, 0, 0, 6, 4, -6, 4, 0, 0, -4, 0, 12, 2, 0, 8, 0, 0, 0, 0, 6, 8, -4, 0, 18, 0, -4, 0, 0, 4, 0, 4, 12, 0, 0, 0, 12, 2, -2, 8, 0, 0, -8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 6, 0, -8, 4, 12, 0, 0, 0, 18, 4, 0, 4, 12, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 25 2014

sign

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

			G.f. = 3 + 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 4*q^6 + 6*q^8 + 6*q^9 + 4*q^11 + ...

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

Cf. A033715, A033761, A112603, A113411, A226240.

A := Basis( ModularForms( Gamma1(32), 1), 81);  3*A[1] + 2*A[2] - 2*A[3] + 4*A[4] + 6*A[5] - 4*A[7] + 6*A[9] + 6*A[10] + 4*A[12] + 12*A[13] + 4*A[16];

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

{a(n) = my(A, p, e); if( n<1, 3*(n==0), A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, if( e>1, 3, -1), p%8>3, (1 + (-1)^e) / 2, e+1)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^3 / (eta(x + A)^2 * eta(x^8 + A)^2) + 2 * eta(x^2 + A)^2 * eta(x^8 + A)^5 / (eta(x^4 + A)^3 * eta(x^16 + A)^2), n))};

a(n) = 2 * b(n) where b(n) is multiplicative with b(2) = -1, b(2^e) = 3 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226240.
a(2*n + 1) = 2 * A113411(n). a(4*n) = 3 * A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n = 3) = 4 * A033761(n). a(8*n + 5) = a(8*n = 7) = 0.

A320069 Expansion of 1/(theta_3(q) * theta_3(q^2)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -2, 2, -4, 10, -16, 20, -32, 58, -86, 112, -164, 260, -368, 480, -672, 986, -1348, 1750, -2372, 3312, -4416, 5684, -7520, 10148, -13266, 16912, -21960, 28896, -37168, 46944, -60032, 77466, -98312, 123076, -155392, 197422, -247696, 307540, -384096, 481776, -598500

Offset: 0

Seiichi Manyama, Oct 05 2018

sign

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

Cf. A000122, A033715, A320068.

CoefficientList[Series[1/Product[EllipticTheta[3, 0, q^k], {k, 1, 2}], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(1/prod(k=1,2, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A033715.

a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (8 * n^(5/4)). - Vaclav Kotesovec, Oct 05 2018

A028572 Expansion of theta_3(z)theta_3(2z) + theta_2(z)theta_2(2z) in powers of q^(1/4).

Original entry on oeis.org

1, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

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

			1 + 4*x^3 + 2*x^4 + 2*x^8 + 4*x^11 + 4*x^12 + 2*x^16 + 4*x^19 + 4*x^24 + ...
1 + 4*q^(3/4) +2*q +2*q^2 +4*q^(11/4) +4*q^3 +2*q^4 + 4*q^(19/4) +4*q^6 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033715, A033761.

terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(3*(n^2 + m^2) + 2*n*m), {n, -max, max}, {m, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* Jean-François Alcover, Dec 03 2015, using 2nd g.f. *)

{a(n) = if( n<1, n==0, qfrep( [3, 1; 1, 3], n)[n] * 2)} /* Michael Somos, Nov 20 2006 */

{a(n) = if( n<1, n==0, if( n%4==1 || n%4==2, 0, 2 * sumdiv( n, d, kronecker( -2, d))))} /* Michael Somos, Mar 23 2012 */

Expansion of phi(x^4) * phi(x^8) + 4 * x^3 * psi(x^8) * psi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Mar 23 2012
G.f.: Sum_{n,m} x^(3*(n^2 + m^2) + 2*n*m). - Michael Somos, Nov 20 2006
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A033715(n). a(8*n + 3) = 4 * A033761(n). - Michael Somos, Mar 23 2012

A129438 Expansion of (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 6, 2, 0, 2, 4, 0, 0, 0, 0, 5, 0, 2, 0, 0, 0

Offset: 0

Michael Somos, Apr 14 2007

nonn

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

			G.f. = 1 + q + 2*q^3 + 2*q^4 + 2*q^8 + 3*q^9 + 2*q^11 + 4*q^12 + 2*q^16 + ...

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. A033715, A033761, A112603, A113411, A125096.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + EllipticTheta[ 4, 0, q^2] EllipticTheta[ 3, 0, q^4]) / 2, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<1, n==0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])};

Moebius transform is period 32 sequence [1, -1, 1, 2, -1, -1, -1, 0, 1, 1, 1, 2, -1, 1, -1, 0, 1, -1, 1, -2, -1, -1, -1, 0, 1, 1, 1, -2, -1, 1, -1, 0, ...].
a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0.
a(n) = A125096(n) unless n=0. a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n).
a(2*n + 1) = A113411(n). a(4*n) = A033715(n). - Michael Somos, Nov 11 2015

A320248 Expansion of Product_{k=1..24} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4446, 5512, 7002, 8614, 10616, 13292, 16260, 19792, 24496, 29724, 35976, 44062, 52992, 63780, 77296, 92518, 110532, 132848, 158036, 187674, 224066, 264960

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_24) to the equation a_1^2 + 2*a_2^2 + ... + 24*a_24^2 = n.

a(24045) = 45676735553670596752038069309732400 and a(24046) = 45676724028345437854371347712212432. So a(24045) > a(24046).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8), A320241 (m=9), A320242 (m=10), A320246 (m=12), A320247 (m=16), this sequence (m=24).

Cf. A320067.

A374294 a(n) is the smallest positive integer k such that A002325(k) = n.

Original entry on oeis.org

1, 3, 9, 27, 81, 99, 729, 297, 1089, 891, 59049, 1683, 531441, 8019, 9801, 5049, 43046721, 18513, 387420489, 15147, 88209, 649539, 31381059609, 31977, 1185921, 5845851, 314721, 136323, 22876792454961, 166617, 205891132094649, 95931, 7144929, 473513931, 10673289, 351747, 150094635296999121

Offset: 1

Seiichi Manyama, Jul 02 2024

nonn

			   n |  a(n)
-----+-----------------------
   2 |     3.
   3 |     9 = 3^2.
   4 |    27 = 3^3.
   5 |    81 = 3^4.
   6 |    99 = 3^2 * 11.
   7 |   729 = 3^6.
   8 |   297 = 3^3 * 11.
   9 |  1089 = 3^2 * 11^2.
  10 |   891 = 3^4 * 11.
  11 | 59049 = 3^10.
  12 |  1683 = 3^2 * 11 * 17.

Cf. A002325, A033715, A033200, A200977, A374295.

If p is prime, a(p) = 3^(p-1).

a(n) is divisible by 3 for n > 1.

A306518 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{d|k} theta_3(q^d).

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 4, 2, 1, 2, 2, 2, 2, 0, 1, 2, 0, 4, 6, 0, 0, 1, 2, 2, 0, 4, 0, 4, 0, 1, 2, 0, 6, 2, 4, 0, 0, 0, 1, 2, 2, 0, 6, 2, 8, 4, 2, 2, 1, 2, 0, 4, 2, 4, 4, 8, 0, 6, 0, 1, 2, 2, 2, 4, 0, 14, 0, 6, 2, 0, 0, 1, 2, 0, 4, 6, 4, 0, 8, 0, 6, 0, 4, 0, 1, 2, 2, 0, 2, 0, 8, 2, 6, 6, 8, 0, 4, 0

Offset: 0

Ilya Gutkovskiy, Feb 21 2019

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  0,  2,  0,  2,  0,  2,  ...
  0,  4,  2,  4,  0,  6,  ...
  2,  2,  6,  4,  2,  6,  ...
  0,  0,  0,  4,  2,  4,  ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=1..48 give A000122, A033715, A033716, A033717, A033718, A033712, A033719, A033720, A033721, A033722, A033723, A033724, A033725, A033726, A033727, A033728, A033729, A033730, A033731, A033732, A033733, A033734, A033735, A033736, A033737, A033738, A033739, A033740, A033741, A033742, A033743, A033744, A033745, A033746, A033747, A033748, A033749, A033750, A033751, A033752, A033753, A033754, A033755, A033756, A033757, A033758, A033759, A033760.

Cf. A320305 (diagonal).

Table[Function[k, SeriesCoefficient[Product[EllipticTheta[3, 0, q^d], {d, Divisors[k]}], {q, 0, n}]][i - n + 1], {i, 0, 13}, {n, 0, i}] // Flatten

G.f. of column k: Product_{d|k} theta_3(q^d).

cp's OEIS Frontend

A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

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A226240 Expansion of phi(q^4) * phi(q^8) + 2 * q *phi(q^2) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

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A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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Sage

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A245572 Expansion of phi(q) * phi(q^2) + 2 * phi(-q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.

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PARI

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A320069 Expansion of 1/(theta_3(q) * theta_3(q^2)), where theta_3() is the Jacobi theta function.

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A028572 Expansion of theta_3(z)*theta_3(2z) + theta_2(z)*theta_2(2z) in powers of q^(1/4).

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A226240 Expansion of phi(q^4) * phi(q^8) + 2 * q phi(q^2) psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

A028572 Expansion of theta_3(z)theta_3(2z) + theta_2(z)theta_2(2z) in powers of q^(1/4).