A035178 -id:A035178 - cp's OEIS frontend

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

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

Offset: 0

N. J. A. Sloane

nonn

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
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 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael D. Hirschhorn, Three classical results on representations of a number, Sem. Lotharingien de Combinat. S42 (1999), B42f.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions, 2010.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
From Michael Somos, Sep 18 2004: (Start)
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).
6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Corrected by Charles R Greathouse IV, Sep 02 2009

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

A279060 Number of divisors of n of the form 6*k + 1.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 05 2016

Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.

			a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.

Cf. A001620, A016629, A222457 (psi(1/6)).

nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale, Apr 27 2018 *)

A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ Antti Karttunen, Jul 09 2017

from sympy import divisors
def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # David Radcliffe, Jun 19 2025

G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A035218(n) - A319995(n). (End)
a(n) = (A035218(n) + A035178(n)) / 2. - David Radcliffe, Jun 19 2025
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

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

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

A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 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, Jan 29 2008

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

Cf. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.

Cf. A035178, A093829, A113447 are same up to sign.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - 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))) / 3, n))}; /* Michael Somos, May 06 2015 */

Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless 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} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( 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 * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). 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 * A131963(n). a(24*n + 19) = 2 * A131964(n).

A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, May 06 2015

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A035178, A248897.

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

a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};

{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==2, 1, p==3, 2, p%6==1, e+1, 1-e%2)))};

Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.
Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if 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^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.
a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 22 2023

A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM 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

Offset: 1

Michael Somos, Sep 04 2015

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

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

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

Cf. A033687, A033762, A093829, A097195, A035178 (apparently gives the absolute values).

Cf. A004016, A005882, A005928.

A004016[q_] := (QPochhammer[q]^3 + 9*q*QPochhammer[q^9]^3)/ QPochhammer[q^3]; A261884[n_] := SeriesCoefficient[(A004016[q] - A004016[q^2] - 2*A004016[q^3] + 2*A004016[q^6])/6, {q, 0, n}]; Table[A261884[n], {n, 1, 50}] (* G. C. Greubel, Sep 24 2017 *)

{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==2, (-1)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) - x * eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};

Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 if e>0, 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} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
a(n) = A093829(n) unless n == 0 (mod 3). a(2*n) = - a(n). a(3*n + 1) = A033687(n).
a(6*n + 1) = A097195(n). a(6*n + 3) = - A033762(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - Amiram Eldar, Nov 23 2023

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 8, 9, 8, 10, 12, 14, 16, 12, 16, 16, 18, 20, 16, 24, 20, 22, 24, 21, 28, 27, 32, 28, 24, 32, 32, 30, 32, 32, 36, 38, 40, 42, 32, 40, 48, 44, 40, 36, 44, 46, 48, 57, 42, 48, 56, 52, 54, 40, 64, 60, 56, 58, 48, 62, 64, 72, 64, 56, 60

Offset: 1

Ilya Gutkovskiy, Sep 12 2019

			G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ... - _Michael Somos_, Oct 23 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586 (fixed points), A035178, A050469, A122373, A326401.

Cf. A175646, A340578.

nmax = 66; CoefficientList[Series[Sum[k x^k (1 + x^(2 k))/(1 + x^(2 k) + x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 6]] &] - DivisorSum[n, # &, MemberQ[{5}, Mod[n/#, 6]] &], {n, 1, 66}]
f[p_, e_] := Which[p < 5, p^e, Mod[p, 6] == 5, (p^(e + 1) - (-1)^(e + 1))/(p + 1), Mod[p, 6] == 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = { sumdiv(n, d, d*((n/d%6==1)-(n/d%6==5))) } \\ Andrew Howroyd, Sep 12 2019

{a(n) = if( n<0, 0, sumdiv( n, d, n/d * kronecker( -12, d)))}; /* Michael Somos, Oct 23 2019 */

a(n) = Sum_{d|n, n/d==1 (mod 6)} d - Sum_{d|n, n/d==5 (mod 6)} d.
G.f.: Sum_{k>=0}  x^(6*k+1) / (1 - x^(6*k+1))^2 - x^(6*k+5) / (1 - x^(6*k+5))^2. - Michael Somos, Oct 23 2019
Multiplicative with a(p^e) = p^e if p < 5, (p^(e+1)-(-1)^(e+1))/(p+1) if p == 5 (mod 6), and (p^(e+1)-1)/(p-1) if p == 1 (mod 6). - Amiram Eldar, Dec 02 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 5 (mod 6)} 1/(1+1/p^2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) = A340578 * A175646 / 2 = 0.48831400806... . - Amiram Eldar, Nov 06 2022

cp's OEIS Frontend

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

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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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A279060 Number of divisors of n of the form 6*k + 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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A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

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A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).

A035246 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -12.