A098098 -id:A098098 - cp's OEIS frontend

A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

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

Offset: 0

Michael Somos, Jul 30 2006

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

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^5 + q^17 + q^29 + q^41 + q^53 + 2*q^65 + q^89 + q^101 + q^113 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
K. Saito, "Extended Affine Root Systems. V. Elliptic Eta-Products and Their Dirichlet Series", Proceedings on Moonshine and related topics (Montreal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. MR1881609 (2003d:11066) See page 215.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A098098, A121363, A258210, A258291, A258832.

a[ n_] := If[ n < 0, 0, Sum[ I^d, {d, Divisors[12 n + 5]}] / (2 I)]; (* Michael Somos, Jul 25 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(3/8) QPochhammer[ x^6]^3 / (QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := Length @ FindInstance[ 24 n + 10 == (6 j + 3)^2 + (6 k + 1)^2 && j >= 0, {j, k}, Integers, 10^9]; (* Michael Somos, Jul 02 2015 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 0, 0, Sum[ Boole[ Mod[d, 4] == 1] - Boole[ Mod[d, 4] == 3], {d, Divisors[12 n + 5]}] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

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

{a(n) = if( n<0, 0, n = 12*n + 5; sumdiv(n, d, (d%4==1) - (d%4==3)) / 2)};

Expansion of f(-x^3) * f(-x^6) / chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q) in powers of q.
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258210.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)).
-2 * a(n) = A121363(3*n + 1).
Convolution square is A098098.
a(n) = (-1)^n * A258832(n) = A052343(3*n + 1). -a(n) = A258291(3*n + 1). 2 * a(n) = A008441(3*n + 1). - Michael Somos, Jul 02 2015
From Peter Bala, Jan 07 2021: (Start)
G.f. A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(12*n + 5)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(12*n + 5)). Cf. A033761. (End)

A121443 Sum of divisors d of n which are odd and n/d is not divisible by 3.

Original entry on oeis.org

1, 1, 3, 1, 6, 3, 8, 1, 9, 6, 12, 3, 14, 8, 18, 1, 18, 9, 20, 6, 24, 12, 24, 3, 31, 14, 27, 8, 30, 18, 32, 1, 36, 18, 48, 9, 38, 20, 42, 6, 42, 24, 44, 12, 54, 24, 48, 3, 57, 31, 54, 14, 54, 27, 72, 8, 60, 30, 60, 18, 62, 32, 72, 1, 84, 36, 68, 18, 72, 48, 72, 9, 74, 38, 93, 20, 96, 42

Offset: 1

Michael Somos, Jul 30 2006, Apr 18 2007

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

			G.f. = q + q^2 + 3*q^3 + q^4 + 6*q^5 + 3*q^6 + 8*q^7 + q^8 + 9*q^9 + 6*q^10 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 86, Eq. (33.124).

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

Cf. A004016, A005882, A005928, A098098, A185717, A291050.

A := Basis( ModularForms( Gamma0(6), 2), 80); A[2] + A[3]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d Mod[ d, 2] Boole[ Mod[ n/d, 3] > 0], {d, Divisors @n}]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]), {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
f[p_, e_] := Which[p == 2, 1, p == 3, p^e, p > 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)

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

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

A = ModularForms( Gamma0(6), 2, prec=80) . basis(); A[1] + A[2]; # Michael Somos, Jun 12 2014

Expansion of c(q) * c(q^2) / 9 where c(q) is a cubic AGM theta function.
Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, -4, ...].
Expansion of (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2)) in powers of q.
Multiplicative with a(2^e) = 1, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 - u*w * (u-2*v) * (v-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) = u1^3*u6 + 2*u2^3*u3 + 3*u2^2*u3^2 + 6*u1*u2*u3*u6 + 48*u2^2*u6^2 - 3*u1^2*u2*u6 - 3*u1*u2*u3^2 - 24*u2^2*u3*u6 - 30*u1*u2*u6^2. - Michael Somos, Apr 18 2007
G.f.: x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(6*k)))^3 / ((1 - x^k) * (1 - x^(2*k))) = Sum_{k>0} k * x^k * (1 - x^k) / (1 + x^(3*k)).
a(2*n) = a(n), a(2*n + 1) = A185717(n). a(3*n) = 3*a(n). a(6*n + 5) = 6 * A098098(n).
G.f.: Sum_{n = -inf..inf} (-1)^n*x^(3*n+1)/(1 - x^(3*n+1))^2. Cf. A124340. - Peter Bala, Jan 06 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/27 = 0.3655409... (A291050). - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)-3^(-s)+2^(1-s)*3^(-s)). - Amiram Eldar, Jan 03 2023

A134077 Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 6, 8, -23, 12, 14, -30, 18, 20, -40, 24, 31, -77, 30, 32, -60, 48, 38, -70, 42, 44, -138, 48, 57, -90, 54, 72, -100, 60, 62, -184, 84, 68, -120, 72, 74, -155, 96, 80, -239, 84, 108, -150, 90, 112, -160, 120, 98, -276, 102, 104, -240, 108, 110, -190, 114

Offset: 0

Michael Somos, Oct 06 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 - 5*x + 6*x^2 + 8*x^3 - 23*x^4 + 12*x^5 + 14*x^6 - 30*x^7 + 18*x^8 + ...
G.f. = q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...

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. A000203, A098098, A118271, A124449, A131944.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x^3, x^3]^3, {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
a[ n_] := SeriesCoefficient[ (1/16) x^(-1/2) (EllipticTheta[ 2, 0, x^(1/2)]^4 - 9 EllipticTheta[ 2, 0, x^(3/2)]^4), {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)

{a(n) = my(A); if ( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^5 * eta(x^6 + A)^3 / ( eta(x^2 + A) * eta(x^3 + A)^3 ), n))};

Expansion of psi(x)^4 - 9 * x * psi(x^3)^4 in powers of x where psi() is a Ramanujan theta function.
Expansion of x^(-1/2) * (b(x)^3 * c(x^2)^2 / (3 * c(x)))^(1/2) in powers of x where b(), c() are cubic AGM functions.
Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-5, -4, -2, -4, -5, -4, ...].
a(n) = b(2*n+1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124449
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.
G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ((1 + x) * (1 + x^2)).
G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is the g.f. of A000203.
a(n) = A131944(2*n + 1) = A118271(2*n + 1). a(3*n + 2) = 6 * A098098(n).

A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, -2, 5, -4, 8, -6, 14, -8, 14, -10, 21, -16, 20, -14, 28, -16, 31, -18, 40, -20, 32, -28, 42, -24, 38, -32, 62, -28, 44, -30, 56, -40, 57, -34, 70, -36, 72, -38, 70, -48, 62, -52, 85, -44, 68, -46, 112, -56, 74, -50, 100, -64, 80, -64, 98, -56, 108, -58, 124

Offset: 0

Michael Somos, Oct 06 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 - 2*x + 5*x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 14*x^6 - 8*x^7 + 14*x^8 - ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 14*q^20 - 8*q^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033686, A098098, A134078, A227696, A263773.

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

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* Michael Somos, Feb 19 2015 *)

{a(n) = my(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 )^2, n))};

{a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* Michael Somos, Feb 19 2015 */

Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.
Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.
a(n) = (-1)^n * A033686(n). 18 * a(n) = A134078(3*n + 2).
From Michael Somos, Feb 19 2015: (Start)
a(2*n + 1) = -2 * A098098(n).
Convolution square of A227696. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - Amiram Eldar, Nov 23 2023

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288

Offset: 0

Michael Somos, Feb 12 2011

sign

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).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};

{a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A252650 Expansion of (eta(q) * eta(q^2) * eta(q^3) / eta(q^6))^2 in powers of q.

Original entry on oeis.org

1, -2, -3, 4, 6, 6, -12, -16, -3, 4, 36, 12, -12, -28, -24, 24, 6, 18, -12, -40, -18, 32, 72, 24, -12, -62, -42, 4, 48, 30, -72, -64, -3, 48, 108, 48, -12, -76, -60, 56, 36, 42, -96, -88, -36, 24, 144, 48, -12, -114, -93, 72, 84, 54, -12, -144, -24, 80, 180

Offset: 0

Michael Somos, Mar 21 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 - 2*q - 3*q^2 + 4*q^3 + 6*q^4 + 6*q^5 - 12*q^6 - 16*q^7 - 3*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
C. Kassel and C. Reutenauer, On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus, arXiv:1505.07229 [math.AG], 2015, see page 31 7.2(d).
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

This is the square of the series in A258210.

Cf. A098098.

A := Basis( ModularForms( Gamma0(36), 2), 58); A[1] - 2*A[2] - 3*A[3] + 4*A[4] + 6*A[5] + 6*A[6] - 12*A[7] - 16*A[8] - 3*A[9] + 4*A[10] + 36*A[11] - 12*A[12];

QP = QPochhammer; s = (QP[q]*QP[q^2]*(QP[q^3]/QP[q^6]))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

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

Expansion of f(-q)^4 * f(q, q^2)^2 / f(-q^3)^2 = f(-q)^4 * f(-q^6)^2 / f(-q, -q^5)^2 in powers of q where f() is a Ramanujan theta function.
Expansion of b(q) * c(q) * sqrt(b(q^2) / (3 * c(q^2))) in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [-2, -4, -4, -4, -2, -4, ...].
G.f.: Product_{k>0} (1 - x^k)^4 / (1 - x^k + x^(2*k))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098098.

A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -4, 5, -8, 7, -8, 9, -10, 14, -12, 16, -14, 15, -20, 17, -18, 19, -24, 26, -22, 23, -28, 25, -32, 32, -28, 29, -30, 38, -32, 33, -40, 40, -44, 42, -38, 39, -40, 57, -42, 43, -44, 45, -62, 47, -56, 49, -56, 62, -52, 53, -60, 64, -68, 64, -58, 59, -60

Offset: 0

Michael Somos, Jun 11 2015

sign

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

			G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...

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. A098098, A121613, A208435, A208457, A258832.

List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # Muniru A Asiru, Jan 30 2018

[(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // Vincenzo Librandi, Jan 30 2018

with(numtheory):
seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # Muniru A Asiru, Jan 30 2018

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* Vincenzo Librandi, Jan 30 2018 *)

{a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};

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

Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
a(n) = (-1)^n * A098098(n) = A208435(2*n + 1) = A208457(2*n + 1). 6 * a(n) = A121613(3*n + 2).
Convolution square of A258832.

A125514 Theta series of 4-dimensional lattice QQF.4.i.

Original entry on oeis.org

1, 4, 20, 4, 52, 24, 20, 32, 116, 4, 120, 48, 52, 56, 160, 24, 244, 72, 20, 80, 312, 32, 240, 96, 116, 124, 280, 4, 416, 120, 120, 128, 500, 48, 360, 192, 52, 152, 400, 56, 696, 168, 160, 176, 624, 24, 480, 192, 244, 228, 620, 72, 728, 216, 20, 288, 928, 80, 600, 240, 312

Offset: 0

N. J. A. Sloane, Jan 31 2007

nonn

			G.f. = 1 + 4*x + 20*x^2 + 4*x^3 + 52*x^4 + 24*x^5 + 20*x^6 + 32*x^7 + 116*x^8 + ...
G.f. = 1 + 4*q^2 + 20*q^4 + 4*q^6 + 52*q^8 + 24*q^10 + 20*q^12 + 32*q^14 + 116*q^16 + ...

John Cannon, Table of n, a(n) for n = 0..5000
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A006353, A030188, A058490, A098098, A123532.

A := Basis(ModularForms( Gamma0(6), 2)); PowerSeries( A[1] + 4*A[2] + 20*A[3], 56); /* Michael Somos, Nov 19 2013 */

a[ n_] := With[{A = QPochhammer[ q] QPochhammer[ q^6], B = QPochhammer[ q^2] QPochhammer[ q^3]}, SeriesCoefficient[ B^7 / A^5 - q A^7 / B^5, {q, 0, n}]] (* Michael Somos, Nov 19 2013 *)

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ], n, 1)[n])} /* Michael Somos, May 27 2012 */

{a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( B^7 / A^5 - x * A^7 / B^5, n))} /* Michael Somos, May 27 2012 */

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 2^(e+2) - 3, if( p==3, 1, (p^(e+1) - 1)/(p - 1))))))} /* Michael Somos, Nov 19 2013 */

A = ModularForms( Gamma0(6), 2, prec=56) . basis(); A[0] + 4*A[1] + 20*A[2]; # Michael Somos, Nov 19 2013

Contribution from Michael Somos, May 27 2012: (Start)
Expansion of (eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = + 5*u^4 + 637*v^4 + 1280*w^4 + 352*u^2*w^2 + 342*u^2*v^2 + 5472*v^2*w^2 + 64*u^3*w + 1024*u*w^3 - 68*u^3*v - 756*u*v^3 - 4352*v*w^3 - 3024*v^3*w - 688*u^2*v*w + 2464*u*v^2*w - 2752*u*v*w^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution of A030188 and A058490. a(3*n) = a(n). (End)
a(n) = 4*b(n) where b(n) is multiplicative and b(2^e) = 2^(e+2) - 3, b(3^e) = 1, b(p^e) = (p^(e+1) - 1)/(p - 1) otherwise. - Michael Somos, Nov 19 2013
a(n) = A006353(n) - A123532(n). a(6*n + 5) = 24 * A098098(n). - Michael Somos, Nov 19 2013

A133739 Expansion of q * (psi(q^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 9, 31, 45, 6, -45, 8, 117, 121, 54, 12, -9, 14, 72, 186, 261, 18, -207, 20, 270, 248, 108, 24, 63, 31, 126, 391, 360, 30, -270, 32, 549, 372, 162, 48, -171, 38, 180, 434, 702, 42, -360, 44, 540, 726, 216, 48, 207, 57, 279, 558, 630, 54, -693, 72, 936, 620

Offset: 1

Michael Somos, Oct 06 2007

sign

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

			G.f. = q + 9*q^2 + 31*q^3 + 45*q^4 + 6*q^5 - 45*q^6 + 8*q^7 + 117*q^8 + ...

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

Cf. A000203, A098098, A134077, A134078.

A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* Michael Somos, Oct 30 2015 */

a[ n_] := SeriesCoefficient[ 2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(3/2)])^3 (EllipticTheta[ 3, 0, x]^5 / EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
QP=QPochhammer; CoefficientList[Series[QP[q^2]^23*QP[q^3]^3*QP[q^12]^6/( QP[q]^9*QP[q^4]^10*QP[q^6]^9), {q,0,50}],q] (* G. C. Greubel, Nov 16 2018 *)

{a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^23 * eta(x^3 + A)^3 * eta(x^12 + A)^6 / (eta(x + A)^9 * eta(x^4 + A)^10 * eta(x^6 + A)^9), n))};

Expansion of eta(q^2)^23 * eta(q^3)^3 * eta(q^12)^6 / (eta(q)^9 * eta(q^4)^10 * eta(q^6)^9) in powers of q.
Euler transform of period 12 sequence [ 9, -14, 6, -4, 9, -8, 9, -4, 6, -14, 9, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134078.
G.f.: f(x) + 6 * f(x^2) + 27 * f(x^3) + 20 * f(x^4) - 162 * f(x^6) + 108 * f(x^12) where f() is the g.f. of A000203.
a(4*n + 2) = 9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

A134078 Expansion of (phi(-q) / phi(-q^2))^3 * phi(q^3)^5 / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 18, -34, 42, -36, 30, -48, 90, -118, 108, -72, 54, -84, 144, -204, 186, -108, 66, -120, 252, -272, 216, -144, 102, -186, 252, -370, 336, -180, 180, -192, 378, -408, 324, -288, 90, -228, 360, -476, 540, -252, 240, -264, 504, -708, 432, -288, 198, -342

Offset: 0

Michael Somos, Oct 06 2007

sign

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

			G.f. = 1 - 6*x + 18*x^2 - 34*x^3 + 42*x^4 - 36*x^5 + 30*x^6 - 48*x^7 + 90*x^8 + ...

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. A098098, A133739, A134079.

a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, -q^2])^3*(EllipticTheta[3, 0, q^3]^5/EllipticTheta[3, 0, -q^6]), {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 22 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^4 + A)^3 * eta(x^6 + A)^23 / ( eta(x^2 + A)^9 * eta(x^3 + A)^10 * eta(x^12 + A)^9 ), n))};

Euler transform of period 12 sequence [ -6, 3, 4, 0, -6, -10, -6, 0, 4, 3, -6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 8 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133739.
a(3*n + 2) = 18 * A134079(n). a(6*n + 5) = -36 * A098098(n).

cp's OEIS Frontend

A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

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A121443 Sum of divisors d of n which are odd and n/d is not divisible by 3.

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A134077 Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

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A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

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A252650 Expansion of (eta(q) * eta(q^2) * eta(q^3) / eta(q^6))^2 in powers of q.

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