A097195 -id:A097195 - cp's OEIS frontend

A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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

Offset: 0

N. J. A. Sloane

Unsigned sequence is expansion of theta series of hexagonal net with respect to a node.
Cubic AGM theta functions: a(q) (see A004016), b(q) (this: A005928), c(q) (A005882).
Denoted by a_3(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - 3*q + 6*q^3 - 3*q^4 - 6*q^7 + 6*q^9 + 6*q^12 - 6*q^13 - 3*q^16 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.34).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
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) see page 695.
J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A005882, A033762, A097195, A113062, A123477, A123530, A246838, A253243.

A := Basis( ModularForms( Gamma1(9), 1), 100); A[1] - 3*A[2] + 6*A[4]; // Michael Somos, Jan 31 2015

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := If[ n < 1, Boole[ n==0], -3 Sum[{1, -1, -3, 1, -1, 3, 1, -1, 0}[[ Mod[ d, 9, 1]]], {d, Divisors @ n}]]; (* Michael Somos, Sep 23 2013 *)

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, if( p%6==1, e+1, !(e%2)))))}; \\ Michael Somos, May 20 2005

{a(n) = my(A = x * O(x^n)); polcoeff( eta(x + A)^3 / eta(x^3 + A), n)}; \\ Michael Somos, May 20 2005

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, -3, 3, 9, -3, 3, -9, -3, 3] [d%9 + 1]))}; \\ Michael Somos, Dec 25 2007

N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(n)-sigma(3*n))*x^n/n));
Vec(gf) \\ Joerg Arndt, Jul 30 2011

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3/eta(q^3))} \\ Altug Alkan, Mar 20 2018

a(n) is the coefficient of q^n in b(q)=eta(q)^3/eta(q^3) = (3/2)*a(q^3)-a(q)/2 where a(q)=theta(Hexagonal). - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
From Michael Somos, May 20 2005: (Start)
Euler transform of period 3 sequence [ -3, -3, -2, ...].
a(n) = -3 * b(n) except for a(0) = 1, where b()=A123477() is multiplicative with b(p^e) = -2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 2, 5 (mod 6).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - 2*u*w^2 + u^2*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) = u1^2*u6 - 2*u1*u2*u6 + 4*u2^2*u6 - 3*u2*u3^2.
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 + u1^2*u3 - 3*u1*u6^2 + u2^2*u3. (End)
a(3*n + 2) = 0. a(3*n + 1) = -A005882(n), a(3*n) = A004016(n). - Michael Somos, Jul 15 2005
a(n) = -3 * A123477(n) unless n=0. |a(n)| = A113062(n).
Moebius transform is period 9 sequence [-3, 3, 9, -3, 3, -9, -3, 3, 0, ...]. - Michael Somos, Dec 25 2007
Expansion of b(q) = a(q^3) - c(q^3) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033687.
G.f.: exp( Sum_{n>=1} (sigma(n)-sigma(3*n))*x^n/n ). - Joerg Arndt, Jul 30 2011
a(n) = (-1)^(mod(n, 3) = 1) * A113062(n). - Michael Somos, Sep 05 2014
a(2*n + 1) = -3 * A123530(n). a(4*n) = a(n). a(4*n + 1) = -3 * A253243(n). a(4*n + 2) = 0. a(4*n + 3) = 6 * A246838(n). a(6*n + 1) = -3 * A097195(n). a(6*n + 3) = 6 * A033762(n). - Michael Somos, Jun 04 2015
G.f.: 1 + Sum_{k>0} -3 * x^k / (1 + x^k + x^(2*k)) + 9 * x^(3*k) / (1 + x^(3*k) + x^(6*k)). - Michael Somos, Jun 04 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

Edited by M. F. Hasler, May 07 2018

A093829 Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.

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, Apr 17 2004

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
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025. See p. 3.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A112300, A113447, A122859.
Cf. A097195, A112604, A112605, A112606, A112607, A112608. - Michael Somos, Sep 27 2013
Cf. A000700, A000122, A010054, A121373.

Basis( ModularForms( Gamma1(6), 1), 90) [2]; /* Michael Somos, Jul 02 2014 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 2, -1, 0} [[ Mod[#, 6, 1]]] &]];
QP = QPochhammer; s = (QP[q]*QP[q^6]^6)/(QP[q^2]^2*QP[q^3]^3) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n) = if( n<1, 0, polcoeff( sum( k=0, n, x^k * (1 - x^k)^2 / (1 + x^(2*k) + x^(4*k)), x * O(x^n)), n))};

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

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d) - if( d%2==0, 2 * kronecker( -3, d/2) ) ))}; /* Michael Somos, May 29 2005 */

ModularForms( Gamma1(6), 1, prec=90).1; # Michael Somos, Sep 27 2013

Expansion of (a(q) - a(q^2)) / 6 = c(q^2)^2 / (3 * c(q)) in powers of q where a(), c() are cubic AGM functions. - Michael Somos, Sep 06 2007
Expansion of (eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [ -1, 1, 2, 1, -1, -2, ...].
Moebius transform is period 6 sequence [ 1, -2, 0, 2, -1, 0, ...] = A112300. - Michael Somos, Jul 16 2006
Multiplicative with a(p^e) = (-1)^e if p=2; a(p^e) = 1 if p=3; a(p^e) = 1+e if p == 1 (mod 6); a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 12^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122859.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 4*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) = u2 * (u2 - u3 - 4*u6) - (u3 + u6) * (u1 - 3*u3 - 3*u6).
G.f.: Sum_{k>0} (x^k - 2 * x^(2*k) + 2 * x^(4*k) - x^(5*k)) / (1 - x^(6*k)) = x * Product_{k>0} ((1 - x^k) * (1 - x^(6*k))^6) / ((1 - x^(2*k))^2 * (1 - x^(3*k))^3).
a(n) = -(-1)^n * A113447(n). - Michael Somos, Jan 31 2015
a(2*n) = -a(n). a(3*n) = a(n). a(6*n + 5) = 0.
A035178(n) = |a(n)|. A033762(n) = a(2*n + 1). A033687(n) = a(3*n + 1).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.302299894039... . - Amiram Eldar, Nov 21 2023

A035178 a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).

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

N. J. A. Sloane

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

			G.f. = q + q^2 + q^3 + q^4 + q^6 + 2*q^7 + q^8 + q^9 + q^12 + 2*q^13 + 2*q^14 + ...

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A093829, A093766, A097197, A107760, A112604, A112605.
Cf. A112606, A112607, A112608, A121361, A123884, A131961, A131962.
Cf. A131963, A131964, A137608.
Cf. A000122, A000700, A010054, A121373.

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

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := If[ n < 1, 0, 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 / EllipticTheta[ 2, 0, q^(3/2)] - 4) / 12, {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
a[n_] := DivisorSum[n, KroneckerSymbol[-12, #]&]; Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d)))}; /* Michael Somos, Apr 18 2004 */

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X))) [n])}; /* Michael Somos, Jun 24 2011 */

{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) - 1) / 3, n))}; /* Michael Somos, Aug 11 2009 */

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%6==5, 1-e%2, 1+e)))}; /* Michael Somos, Aug 04 2015 */

Moebius transform is period 6 sequence [ 1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Feb 14 2006
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) - (u2 -u6) * (1 + 3*u6). - Michael Somos, May 29 2005
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -12, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -12, p) * p^-s)). - Michael Somos, Jun 24 2011
a(n) is multiplicative with a(p^e) = 1 if p=2 or p=3, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) = Sum_{k>=0} x^(6*k + 1) / (1 - x^(6*k + 1)) - x^(6*k + 5) / (1 - x^(6*k + 5)). - Michael Somos, Feb 14 2006
a(n) = |A093829(n)| = -(-1)^n * A137608(n) = a(2*n) = a(3*n).  a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
From Michael Somos, Aug 11 2009: (Start)
3 * a(n) = A107760(n) unless n=0. a(2*n + 1) = A033762(n). a(3*n + 1) = A033687(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 7) = 2 * A112608(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 * A131963(n). a(24*n + 19) = 2 * A131964(n). (End)
Expansion of (psi(q)^3 / psi(q^3) - 1) / 3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 16 2023

Definition edited by Michael Somos, Aug 11 2009

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

Original entry on oeis.org

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

A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 28 2006, Apr 04 2008

Expansion of (a(q) - 2 * a(q^2) - a(q^4) + 2*a(q^8)) / 6 in powers of q where a() is a cubic AGM function.

			q - 2*q^2 + q^3 - 2*q^6 + 2*q^7 + q^9 + 2*q^13 - 4*q^14 - 2*q^18 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

A033762(n) = a(2*n+1). A112604(n) = a(4*n+1). -2 * A033762(n) = a(4*n+2). A112605(n) = a(4*n+3). A097195(n) = a(6*n+1). A112606(n) = a(8*n+1). -2 * A112604(n) = a(8*n+2). A112608(n) = a(8*n+3). 2 * A112607(n) = a(8*n+5). -2 * A112605(n) = a(8*n+6). 2 * A112609(n) = a(8*n+7).
A123884(n) = a(12*n+1). 2 * A121361(n) = a(12*n+7). A131961(n) = a(24*n+1). 2 * A131962(n) = a(24*n+7). A112608(n) = a(24*n+9). 2 * A131963(n) = a(24*n+13). 2 * A131964(n) = a(24*n+19).
Cf. A093766, A136748.

QP = QPochhammer; s = QP[x]^2*QP[x^6]^4*QP[x^8]*(QP[x^24]/(QP[x^2]*QP[x^3]* QP[x^12])^2) + O[x]^105; CoefficientList[s, x] (* Jean-François Alcover, Nov 06 2015, adapted from PARI, updated Dec 06 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, 1, d/2%2*-2)*kronecker(-12, n/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, -2*(e<2), if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))}

Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -2, -2, -1, 0, 0, -2, 0, -2, 0, 0, -1, -2, -2, -2, 0, 0, 0, -2, -2, ...].
Moebius transform is period 24 sequence [ 1, -3, 0, 2, -1, 0, 1, 0, 0, 3, -1, 0, 1, -3, 0, 0, -1, 0, 1, -2, 0, 3, -1, 0, ...].
a(n) is multiplicative with a(2) = -2, a(2^e) = 0 if e>1, 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. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A136748.
G.f.: x * Product_{k>0} (1 -x^(6*k)) * (1 - x^k + x^(2*k))^2 * (1 - x^(8*k)) * (1 + x^(12*k)) / (1 + x^(6*k)).
a(4*n) = a(6*n + 4) = a(6*n + 5) = 0. a(3*n) = a(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 22 2024

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

cp's OEIS Frontend

A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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A093829 Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.

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A035178 a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).

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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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A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

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