A107760 -id:A107760 - cp's OEIS frontend

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

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

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

A132979 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, 6, -12, 24, -45, 78, -132, 222, -363, 576, -900, 1392, -2121, 3180, -4716, 6936, -10098, 14550, -20796, 29520, -41595, 58176, -80856, 111750, -153561, 209820, -285240, 385968, -519840, 696960, -930516, 1237470, -1639314, 2163456, -2845080, 3728904, -4871211

Offset: 0

Michael Somos, Sep 07 2007

sign

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

			G.f. = 1 - 3*q + 6*q^2 - 12*q^3 + 24*q^4 - 45*q^5 + 78*q^6 - 132*q^7 + ...

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. A107760, A132974.

a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)]^3, {q, 0, n}]; (* Michael Somos, Jun 20 2015 *)

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

Expansion of eta(q)^3 * eta(q^6)^2 / ( eta(q^2)^6 * eta(q^3) ) in powers of q.
Euler transform of period 6 sequence [ -3, 3, -2, 3, -3, 2, ...].
G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k)) / ( (1 + x^k) * (1 - x^(2*k)) )^3.
a(n) = (-1)^n * A132974(n). Convolution inverse of A107760.

A253625 Expansion of psi(q^2) * f(-q, q^2)^2 / f(-q, -q^5) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 06 2015

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 - q + 3*q^2 - q^3 + 3*q^4 + 3*q^6 - 2*q^7 + 3*q^8 - q^9 + 3*q^12 + ...

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. A000122, A000700, A004016, A005882, A005928, A010054, A033687, A033762, A093766, A097195, A107760, A112604, A112605, A121361, A121373, A122861, A123884, A253623, A253626.

A := Basis( ModularForms( Gamma1(12), 1), 81); A[1] - A[2] + 3*A[3] - A[4] + 3*A[5];

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

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

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

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

Expansion of psi(q^2)^2 * phi(q^3)^2 / (psi(q) * psi(q^3)) = f(-q) * f(-q^3) * (chi(q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (-a(q) - 3*a(q^2) + 4*a(q^4)) / 6 = b(q^4) * (b(q) + 2*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q) * eta(q^4)^4 * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -1, 3, 2, -1, -1, -2, -1, -1, 2, 3, -1, -2, ...].
Moebius transform is period 12 sequence [ -1, 4, 0, 0, 1, 0, -1, 0, 0, -4, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253623.
a(n) = -b(n) where b() is multiplicative with b(2^e) = -3 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).
G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*4) * (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 + x^(3*k))^4 / (1 - x^(2*k) + x^(4*k))^4.
a(n) = (-1)^n * A253626(n). a(2*n) = A107760(n). a(2*n + 1) = - A033762(n). a(3*n) = a(n). a(3*n + 1) = - A122861(n). a(4*n + 1) = - A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = - A112605(n).
a(6*n + 1) = - A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = - A123884(n). a(12*n + 7) = -2 * A121361(n). a(12*n + 10) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jun 08 2025

Typo in formula fixed by Colin Barker, Jan 08 2015

A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 15, 24, 33, 36, 33, 48, 69, 78, 90, 72, 51, 84, 120, 144, 141, 108, 87, 120, 198, 192, 180, 144, 87, 186, 210, 240, 264, 180, 198, 192, 285, 288, 270, 288, 105, 228, 300, 336, 414, 252, 264, 264, 396, 468, 360, 288, 159, 342, 465, 432, 462, 324, 249

Offset: 0

Michael Somos, Sep 19 2013

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 8 and 32 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

			G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A107760, A227226.

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];

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

A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];

Expansion of (a(q) + a(q^2))^2 / 4 in powers of q where a() is a cubic AGM theta function.
Expansion of (b(q^2)^2 / b(q))^2 in powers of q where b() is a cubic AGM theta function.
Expansion of (eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2))^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.
Convolution square of A107760.
Euler transform of period 6 sequence [6, -6, 4, -6, 6, -4, ...].

A253626 Expansion of psi(q^2) * f(q, q^2)^2 / f(q, q^5) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 06 2015

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

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, A093602, A097195, A107760, A112604, A112605, A121361, A122861, A123884, A253625.

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

A := Basis( ModularForms( Gamma1(12), 1), 86); A[1] + A[2] + 3*A[3] + A[4] + 3*A[5];

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

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-12, d) + if(d%2, 0, 2 * kronecker(-12, d/2))))};

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

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

Expansion of psi(q^2)^2 * phi(-q^3)^2 / (psi(-q) * psi(-q^3)) = f(q) * f(q^3) * (chi(-q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (a(q) + 3*a(q^2) + 2*a(q^4)) / 6 = b(q^4) * (-b(q) + 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^3)^3 * eta(q^4)^3 / ( eta(q) * eta(q^2) * eta(q^6) * eta(q^12) ) in powers of q.
Euler transform of period 12 sequence [ 1, 2, -2, -1, 1, 0, 1, -1, -2, 2, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, 0, -1, 0, 1, 0, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) is multiplicative with a(0) = 1, a(2^e) = 3 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.: 1 + Sum_{k>0} (3 - (k mod 2)*2) * (q^k + q^(3*k)) / (1 + q^(2*k) + q^(4*k)).
G.f.: Product_{k>0} (1 - q^(3*k))^3 * (1 - q^(4*k))^3 / ( (1 - q^k) * (1 - q^(2*k)) * (1 - q^(6*k)) * (1 - q^(12*k)) ).
a(n) = (-1)^n * A253625(n). a(2*n) = A107760(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(4*n + 1) = A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). a(12*n + 10) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Jan 21 2024

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

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

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

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A132979 Expansion of psi(q^3) / psi(q)^3 in powers of q where psi() is a Ramanujan theta function.

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A253625 Expansion of psi(q^2) * f(-q, q^2)^2 / f(-q, -q^5) in powers of q where psi(), f() are Ramanujan theta functions.

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A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

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