A033687 -id:A033687 - cp's OEIS frontend

A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

1, -3, 2, 0, 2, -3, 2, 0, 1, -6, 2, 0, 2, 0, 2, 0, 3, -6, 0, 0, 2, -3, 2, 0, 2, -6, 2, 0, 0, 0, 4, 0, 2, -3, 2, 0, 2, -6, 0, 0, 1, -6, 2, 0, 4, 0, 2, 0, 0, -6, 2, 0, 2, 0, 2, 0, 3, -6, 2, 0, 2, 0, 0, 0, 2, -9, 2, 0, 0, -6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, -3, 4, 0, 0, -6, 2, 0, 2, 0, 2, 0, 1, -6, 0, 0, 4, -6, 2, 0, 2

Offset: 0

Michael Somos, Sep 15 2006

sign

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

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, A097109, A097195, A115979, A129576.

Cf. A000122, A000700, A010054, A121373.

A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

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

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

Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].
a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(4n+3) = 0.
a(n) = A115979(3n+1) = A097109(3n+1).
a(2n) = A097195(n) = A033687(2n); a(2n+1) = -3*A033687(2n+1).
a(n) = (-1)^n * A129576(n). - Amiram Eldar, Jan 28 2024

A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.

Original entry on oeis.org

1, -2, -1, 0, 5, 4, -7, 0, -5, 2, -4, 0, 11, 0, 8, 0, -6, -10, 0, 0, -1, -8, 5, 0, -7, 14, 17, 0, 0, 0, -5, 0, -19, 10, -13, 0, 2, -4, 0, 0, -11, 8, 20, 0, 7, 0, 23, 0, 0, -22, -19, 0, 14, 0, -25, 0, 12, -16, 5, 0, -7, 0, 0, 0, 23, 12, 11, 0, 0, 20, -13, 0, 4, 0, -28, 0, -22, 0, 0, 0, 17, 2, -35, 0, 0, 16, -11, 0, 0, -10

Offset: 0

N. J. A. Sloane

sign

Number 44 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_2(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique weight 2 newform of level 27.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 2*x - x^2 + 5*x^4 + 4*x^5 - 7*x^6 - 5*x^8 + 2*x^9 - 4*x^10 + 11*x^12 + ...
G.f. = q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030203, A279005.

A := Basis( ModularForms( Gamma0(27), 2), 271); A[2] - 2*A[5]; /* Michael Somos, Jun 12 2014 */

qEigenform( EllipticCurve( [0, 0, 1, 0, 0]), 271); /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(27), 2), 271)[1]; /* Michael Somos, Mar 24 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^2, {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(e/2)), for( i=1, sqrtint(4*p\27), if( issquare(4*p - 27*i^2, &y), break)); a0=1; a1 = y*= (-1)^(y%3); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2, n))}; /* Michael Somos, Feb 19 2007 */

{a(n) = ellak( ellinit( [0, 0, 1, 0, 0], 1), 3*n + 1)}; /* Michael Somos, Jun 12 2014 */

ModularForms( Gamma0(27), 2, prec=271).0; # Michael Somos, Jun 12 2014

Expansion of q^(-1/3) * b(q) * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Coefficients of L-series for elliptic curve "27a3": y^2 + y = x^3. - Michael Somos, Aug 13 2006
Euler transform of period 3 sequence [-2, -2, -4, ...]. - Michael Somos, Dec 06 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2.
a(n) = b(3*n + 1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(e/2), if p == 2 (mod 3), otherwise b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)). - Michael Somos, Aug 13 2006
Given g.f. A(x), then B(q)= q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u*w * (u + 4*w). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
a(4*n + 1) = -2 * a(n). - Michael Somos, Dec 06 2004
a(25*n + 8) = -5 * a(n). Convolution square of A030203. - Michael Somos, Mar 13 2012

A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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. = 3*x + 3*x^4 + 6*x^7 + 6*x^13 + 3*x^16 + 6*x^19 + 3*x^25 + 6*x^28 + ...
G.f. = 3*q^(1/3) + 3*q^(4/3) + 6*q^(7/3) + 6*q^(13/3) + 3*q^(16/3) + 6*q^(19/3) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004016, A005882, A005928, A033687, A045833, A217219, A248897.

Cf. A000122, A000700, A010054, A121373.

a[n_] := If[Mod[n, 3] != 1, 0, 3*DivisorSum[n, KroneckerSymbol[#, 3]&]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 03 2015, adapted from PARI *)
s = 3q*(QPochhammer[q^9]^3/QPochhammer[q^3])+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

{a(n) = if( (n<0) || (n%3 != 1), 0, 3 * sumdiv( n, d, kronecker( d, 3)))}; \\ Michael Somos, Jul 16 2005

{a(n) = my(A); if( (n<0) || (n%3 != 1), 0, n = n\3; A = x * O(x^n); 3 * polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, Jul 16 2005

a(3*n) = a(3*n + 2) = 0.
a(3*n + 1) = A005882(n) = 3 * A033687(n) = -A005928(3*n + 1) = A004016(3*n + 1) / 2.
Expansion of 3 * eta(q^3)^3 / eta(q) in powers of q^(1/3).
G.f.: 3 * x * Product_{k>0} (1 - x^(9*k))^3 / (1 - x^(3*k)) = 3 * Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(4*k)) / (1 - x^(9*k)). - Michael Somos, Jul 15 2005
Expansion of c(x^3) in powers of x where c(x) is a cubic AGM theta function. - Michael Somos, Oct 17 2006
From Michael Somos, Dec 25 2011: (Start)
G.f.: Sum_{i, j in Z} x^(3 * (i^2 + i*j + j^2 + i + j) + 1).
G.f.: Sum_{i, j, k} x^(3 * Q(i, j, k) - 2) where Q(i, j, k) = i*i + j*j + k*k + i*j + i*k + j*k and the sum is over all integer i, j, k where i + j + k = 1. (End)
a(n) = A217219(n)/2. - N. J. A. Sloane, Oct 05 2012
Expansion of 2 * x * psi(x^6) * f(x^6, x^12) + x * phi(x^3) * f(x^3, x^15) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 09 2018
From Amiram Eldar, Oct 13 2022: (Start)
a(n) = 3*A045833(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)

A144614 Sum of divisors of 3*n + 1.

Original entry on oeis.org

1, 7, 8, 18, 14, 31, 20, 36, 31, 56, 32, 54, 38, 90, 44, 72, 57, 98, 72, 90, 62, 127, 68, 144, 74, 140, 80, 126, 108, 180, 112, 144, 98, 217, 104, 162, 110, 248, 144, 180, 133, 224, 128, 252, 160, 270, 140, 216, 180, 266, 152, 288, 158, 378, 164, 252, 183, 308

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. = 1 + 7*x + 8*x^2 + 18*x^3 + 14*x^4 + 31*x^5 + 20*x^6 + 36*x^7 + 31*x^8 + 56*x^9 +...
G.f. = q + 7*q^4 + 8*q^7 + 18*q^10 + 14*q^13 + 31*q^16 + 20*q^19 + 36*q^22 + 31*q^25 + ...

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000203, A004016, A005882, A005928, A016777, A033687.

Basis( ModularForms( Gamma0(9), 2), 173)[2]; /* Michael Somos, Jun 10 2015 */

a[ n_] := If[ n < 0, 0, DivisorSigma[1, 3 n + 1]]; (* Michael Somos, May 26 2014 *)
DivisorSigma[1,3*Range[0,60]+1] (* Harvey P. Dale, Mar 20 2023 *)

{a(n) = if( n<0, 0, sigma( 3*n + 1))}; /* Michael Somos, May 30 2012 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) * eta(x^3 + A)^2 / eta(x + A), n))}; /* Michael Somos, Jun 09 2012 */

ModularForms( Gamma0(9), 2, prec=70).1; # _Michael Somos, May 26 2014 */

Expansion of q^(-1/3) * a(q) * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 30 2012
From Michael Somos, Jun 09 2012: (Start)
Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3) * eta(q^3)^2 / eta(q) in powers of q.
a(n) = A000203(3*n + 1).
Convolution of A004016 and A033687. (End)
Sum_{k=1..n} a(k) = (2*Pi^2/9) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

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

Original entry on oeis.org

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

A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 12, -6, -6, 0, 12, -12, 12, -6, 0, 0, -6, -12, 24, 0, -6, 0, 12, -12, 0, -12, 0, 0, 12, -6, 24, -6, -12, 0, 0, -12, 12, 0, 0, 0, -6, -12, 24, -12, 0, 0, 24, -12, 0, 0, 0, 0, -6, -18, 12, 0, -12, 0, 12, 0, 24, -12, 0, 0, 0, -12, 24, -12, -6, 0, 0, -12, 0, 0, 0, 0, 12, -12, 24, -6, -12, 0, 24, -12, 0, -6, 0, 0, -12, 0, 24

Offset: 0

Michael Somos, Sep 15 2006

sign

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

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

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

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, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A113660, A121361, A122860, A123884, A227354.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 / EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Sep 27 2013 *)

{a(n)= if( n<1, n==0, 6 * sumdiv(n, d, (-1)^(n/d) * kronecker( -3, d)))}

{a(n)= if( n<1, n==0, -6 * sumdiv(n, d, (2 + (-1)^d) * kronecker( -3, d)))}

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

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

Expansion of 2*a(q^2) - a(q) = b(q)^2 / b(q^2) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [ -6, -3, -4, -3, -6, -2, ...].
Moebius transform is period 6 sequence [ -6, 18, 0, -18, 6, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v*(u+v)^2 - 2*u*w*(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-u3+u6) * (u1+2*u2+u3) - (2*u1+u2-2*u3-u6) * (u1+2*u2-u3).
G.f.: Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3k)) = 1 + 6 * Sum_{k>0} (-1)^k * x^k / (1 + x^k + x^(2*k)).
G.f.: 1 - 6 * (Sum_{k>0} x^(3*k - 2) / (1 + x^(3*k - 2)) - x(3*k - 1)
/ (1 + x^(3*k - 1))).
a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.
(-1)^n * a(n) = A113660(n). -6 * a(n) = A122860(n) if n>0.
a(2*n) = A227354(n). a(2*n + 1) = -6 * A033762(n). a(3*n + 1) = -6 * A033687(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 3) = -6 * A112605(n). a(6*n + 1) = -6 * A097195(n). a(8*n + 1) = -6 * A112606(n). a(8*n + 3) = -6 * A112608(n). a(8*n + 5) = -12 * A112607(n-1). a(8*n + 7) = -12 * A112609(n). a(12*n + 1) = -6 * A123884(n). a(12*n + 7) = -12 * A121361(n). - Michael Somos, Sep 27 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - Amiram Eldar, Nov 23 2023

A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 2, 1, 1, 0, 0, -1, 2, 2, 0, -1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, -2, 0, 0, 2, 1, 0, 0, 0, -1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, 1, 0, -2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, -1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, -2, 0, 2, 2, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, -1, 0, 0, 2, 2, 0

Offset: 1

Michael Somos, Nov 02 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. = q + q^2 + q^3 - q^4 + q^6 + 2*q^7 + q^8 + q^9 - q^12 + 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Li-Chien Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1105, equation (3.8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A073010, A093829, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964. A133637.

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

A := Basis( ModularForms( Gamma1(24), 1), 106); A[2] + A[3] + A[4] - A[5] + A[7] + 2*A[8] + A[9] + A[10]; /* Michael Somos, May 07 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, Jan 31 2015 *)

{a(n) = if( n<1, 0, -(-1)^max( 1, valuation( n, 2)) * sumdiv(n, d, kronecker( -12, d)))};

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

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3), n))};

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 0, -2, -1, 0, 1, 2, 0, 0,-1][d%12 + 1]))}; /* Michael Somos, May 07 2015 */

Expansion of (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, 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).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - 2 * x^(12*k - 8) / (1 - x^(12*k - 8)) + 2 * x^(12*k - 4) / (1 - x^(12*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(3*k))^2 / (1 + x^(4*k) + x^(8*k)).
G.f.: x * Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)) * ((1 - x^(12*k - 6)) / (1 - x^(3*k)))^3.
Expansion of theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132973.
a(n) = -(-1)^n * A093829(n). - Michael Somos, Jan 31 2015
Convolution inverse of A133637.
a(3*n) = a(n). a(6*n + 5) = a(12*n + 10) = 0. |a(n)| = A035178(n).
a(2*n) = A093829(n). a(2*n + 1) = A033762(n).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 6) = A112605(n). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A121963(n). a(24*n + 19) = 2 * A131964(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 23 2023

A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 26 2006

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

			G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009

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, A097195, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.

Cf. A000700, A000122, A010054, A121373.

QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

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

{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), n))}

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # Michael Somos, Sep 27 2013

Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.
a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).
G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.
G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).
From Michael Somos, Aug 11 2009: (Start)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.
a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.
a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Nov 14 2023

A132973 Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 07 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 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*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, A097195, A107760, A132973, A132974, A227696.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]/2, {q, 0, n}]; (* Michael Somos, May 26 2013 *)

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

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

Expansion of b(q^2)^2 / b(-q) = b(q) * b(q^4) / b(q^2) in powers of q where b() is a cubic AGM theta function.
Expansion of (a(q^2) + 2 * a(q^4) - a(q)) / 2 = (c(q)^2 - 5 * c(q) * c(q^4) + 4 * c(q^4)^2) / (3 * c(q^2)) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 26 2013
Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, -2, 0, -3, -2, ...].
Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A113447.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * (1 + x^(6*k))).
G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: 1 + 3 * ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = (-1)^n * A107760(n). Convolution inverse of A132974.
a(2*n) = A107760(n). a(2*n + 1) = -3 * A033762(n). a(3*n) = A132973(n). a(3*n + 1) = -3 * A227696(n). - Michael Somos, Oct 31 2015
a(6*n + 1) = -3 * A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. - Michael Somos, Oct 31 2015

A227696 Expansion of f(x^3)^3 / f(x) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 2, 0, 2, -1, 2, 0, 1, -2, 2, 0, 2, 0, 2, 0, 3, -2, 0, 0, 2, -1, 2, 0, 2, -2, 2, 0, 0, 0, 4, 0, 2, -1, 2, 0, 2, -2, 0, 0, 1, -2, 2, 0, 4, 0, 2, 0, 0, -2, 2, 0, 2, 0, 2, 0, 3, -2, 2, 0, 2, 0, 0, 0, 2, -3, 2, 0, 0, -2, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, -2

Offset: 0

Michael Somos, Sep 22 2013

sign

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

			G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 2*x^6 + x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 2*q^19 + q^25 - 2*q^28 + 2*q^31 + ...

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, A097195, A121361, A123884, A226535.

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

{a(n) = local(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, n))}

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^9 / (eta(q^2) * eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -4, -1, 1, 2, 2, -1, -2, ...].
Moebius transform is period 36 sequence [ 1, -1, -1, -1, -1, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 1, -1, 0, -1, -1, -1, 1, 1, 1, 1, -1, 0, ...].
a(n) = b(3*n+1) where b(n) is multiplicative with b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^3) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3)^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A226535.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^k).
a(n) = (-1)^n * A033687(n). a(4*n + 3) = 0.
a(2*n) = A097195(n). a(4*n) = A123884(n). a(4*n + 1) = - A033687(n). a(4*n + 2) = 2 * A121361(n).

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A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

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A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.

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A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.

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A144614 Sum of divisors of 3*n + 1.

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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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A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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