A030203 -id:A030203 - cp's OEIS frontend

A030207 Expansion of eta(q)^2 * eta(q^2) * eta(q^4) * eta(q^8)^2 in powers of q.

1, -2, -2, 4, 0, 4, 0, -8, -5, 0, 14, -8, 0, 0, 0, 16, 2, 10, -34, 0, 0, -28, 0, 16, 25, 0, 28, 0, 0, 0, 0, -32, -28, -4, 0, -20, 0, 68, 0, 0, -46, 0, 14, 56, 0, 0, 0, -32, 49, -50, -4, 0, 0, -56, 0, 0, 68, 0, -82, 0, 0, 0, 0, 64, 0, 56, 62, 8, 0, 0, 0, 40, -142, 0, -50, -136, 0, 0, 0, 0, -11, 92, 158, 0, 0, -28, 0

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Unique cusp form of weight 3 for congruence group Gamma_1(8). - Michael Somos, Aug 11 2011
Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2.
For n nonzero, a(n) is nonzero if and only if n is in A002479.
Number 20 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 2*q^2 - 2*q^3 + 4*q^4 + 4*q^6 - 8*q^8 - 5*q^9 + 14*q^11 - 8*q^12 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A128712, A128713.

Basis( CuspForms( Gamma1(8), 3), 100) [1]; /* Michael Somos, May 27 2014 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 3, 0, q^2] EllipticTheta[ 2, 0, q^2]^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 3, 0, q]^5 EllipticTheta[ 3, 0, q^2] + 3 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 3, 0, q^2]^3 - 2 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^5) / 4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * eta(x^2 + A) * eta(x^4 + A), n))}; /* Michael Somos, May 28 2007 */

{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, p%8>4, if( e%2, 0, p^e), for( x=1, sqrtint(p\2), if( issquare( p - 2*x^2, &y), break)); y = 4*y^2 - 2*p; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 13 2007 */

CuspForms( Gamma1(8), 3, prec = 100).0; # Michael Somos, Aug 11 2011

Expansion of q * phi(q) * phi(-q)^2 * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, May 28 2007
Expansion of (3 * phi(q)^3 * phi(q^2)^3 - 2 * phi(q) * phi(q^2)^5 - phi(q)^5 * phi(q^2)) / 4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 13 2007
Euler transform of period 8 sequence [-2, -3, -2, -4, -2, -3, -2, -6, ...]. - Michael Somos, May 28 2007
a(n) is multiplicative with a(2^e) = (-2)^e, a(p^e) = (1+(-1)^e)/2 * p^e if p == 5, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3 (mod 8) where a(p) = 4*x^2 -2*p and p = x^2 + 2*y^2. - Michael Somos, Jun 13 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007
G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + 2*v*v). - Michael Somos, Jun 14 2007
G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 + x^(2*k))^3 * (1 + x^(4*k))^6. - Michael Somos, May 28 2007
a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = -2*a(n). a(8*n + 1) = A128712(n). a(8*n + 3) = -2 * A128713(n).

A030209 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, -16, -8, 9, -12, 12, -12, 38, 32, -18, 16, -126, -18, 20, 24, 48, -24, 168, 24, -89, -76, -27, -64, 30, 36, -88, -32, -36, 252, -96, 36, 254, -40, -114, -48, 42, -96, -52, 48, 54, -336, -96, -48, -87, 178, 378, 152, 198, 54, 72, 128

Offset: 1

N. J. A. Sloane

Identical to table 1, p. 493, of Alaca citation. - Jonathan Vos Post, May 24 2007
Unique cusp form of weight 4 for congruence group Gamma_1(6). - Michael Somos, Aug 11 2011
Number 14 of the 74 eta-quotients listed in Table I of Martin (1996).
The table 1, p. 493 of Alaca reference is the first 50 values of c_6(n). - Michael Somos, May 17 2015

			G.f. = q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + 6*q^6 - 16*q^7 - 8*q^8 + 9*q^9 - 12*q^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Saban Alaca and Kenneth Williams, Evaluation of the convolution sums sum_{l+6m=n} sigma(l)*sigma(m) and sum_{2l+3m=n} sigma(l)*sigma(m), J. Number Theory 124 (2007), no. 2, 491-510. MR2321376 (2008a:11005)
K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.
Brian Conrey and David Farmer (?), Eta Products and Quotients which are Newforms.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 6 and weight 4.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030188, A181101, A186100.

Basis( CuspForms( Gamma1(6), 4), 57) [1]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A))^2, n))}; /* Michael Somos, Feb 14 2006 */

CuspForms( Gamma1(6), 4, prec = 57).0; # Michael Somos, Aug 11 2011

Euler transform of period 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - Michael Somos, Feb 13 2006
a(n) is multiplicative with a(p^e) = (-p)^e if p<5, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.
a(2*n) = -2 * a(n). Convolution square of A030188. - Michael Somos, May 27 2012
Convolution with A181102 is A186100. - Michael Somos, Jul 07 2015

A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

Original entry on oeis.org

1, 3, 9, 13, 24, 27, 50, 51, 81, 72, 120, 117, 170, 150, 216, 205, 288, 243, 362, 312, 450, 360, 528, 459, 601, 510, 729, 650, 840, 648, 962, 819, 1080, 864, 1200, 1053, 1370, 1086, 1530, 1224, 1680, 1350, 1850, 1560, 1944, 1584, 2208, 1845, 2451, 1803, 2592

Offset: 1

Michael Somos, May 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 3 of the 74 eta-quotients listed in Table I of Martin (1996).
a(n+1) is the number of partition triples of n where each partition is 3-core (see Theorem 3.1 of Wang link).
Convolution cube of A033687.
Convolution square is A198958. - Michael Somos, Dec 26 2015

			G.f. = q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 + ...

George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001). See p. 314, Eq. (14.2.14).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vaclav Kotesovec)
Jonathan M. Borwein and Peter 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 697.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A004016, A005882, A005928, A033687, A109041, A129404, A198958, A231947.

A := Basis( ModularForms( Gamma1(3), 3), 52); A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, #^2 KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3]^3 / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^9 / (1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

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

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

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

a(n) = sumdiv(n, d, ((d % 3) == 1)*(n/d)^2) - sumdiv(n, d, ((d % 3)== 2)*(n/d)^2); \\ Michel Marcus, Jul 14 2015

Expansion of (c(q) / 3)^3 in powers of q where c(q) is a cubic AGM theta function.
Euler transform of period 3 sequence [ 3, 3, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 6*u*v*w + 8*u*w^2 - u^2*w.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) = x * Product_{k>0} (1 - x^(3*k))^9 / (1 - x^k)^3.
a(n) is multiplicative and a(p^e) = ((p^2)^(e+1) - u^(e+1)) / (p^2 - u) where u = 0, 1, -1 when p == 0, 1, 2 (mod 3). - Michael Somos, Oct 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109041.
a(3*n) = 9 * a(n). a(3*n + 1) = A231947(n). -  Michael Somos, May 18 2015
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 4*Pi^3/(81*sqrt(3)) = 0.8840238... (A129404). - Amiram Eldar, Nov 09 2023

A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110

Offset: 0

Michael Somos, Aug 10 2006

sign

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

Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179
Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sigma(2*n + 1))};

A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013

Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2))^4 in powers of q.
Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).
Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013
G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.
G.f.: Sum_{k>0} -(-1)^k * (2*k - 1) * x^(k - 1) / (1 + x^(2*k - 1)).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.
G.f.: (Sum_{k>0} (-1)^floor(k/2) * x^((k^2 - k)/2))^4.
G.f.: Sum_{k>0} (-1)^k * (2*k - 1) * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).
Convolution square of A134343. - Michael Somos, Jun 20 2012
a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 12, 12, 14, 12, 12, 16, 16, 18, 18, 16, 18, 24, 24, 24, 21, 28, 27, 24, 28, 24, 30, 32, 36, 32, 24, 36, 38, 36, 42, 32, 40, 36, 42, 48, 36, 48, 48, 48, 43, 42, 48, 56, 52, 54, 48, 48, 54, 56, 60, 48, 62, 60, 54, 64, 56, 72, 66, 64, 72, 48, 72, 72

Offset: 1

Michael Somos, Nov 08 2006

Number 24 of the 74 eta-quotients listed in Table I of Martin (1996).

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

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A258414.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, Sum[ n/d KroneckerSymbol[ 12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^2 QPochhammer[ q^4] QPochhammer[ q^12]/QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

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

{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 12, p); (p^(e+1) - f^(e+1)) / (p - f)))};

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

Expansion of (eta(q^2) * eta(q^3) / eta(q))^2 * eta(q^4) * eta(q^12) in powers of q.
Euler transform of period 12 sequence [ 2, 0, 0, -1, 2, -2, 2, -1, 0, 0, 2, -4, ...].
a(n) is multiplicative with a(p^e) = p^e if p<5, a(p^e) = (p^(e+1) - 1) / (p-1) if p == 1, 11 (mod 12), a(p^e) = (p^(e+1) + (-1)^e) / (p+1) if p == 5, 7 (mod 12).
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 - x^(2*k) + x^(4*k)).
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(4*k)) * (1 - x^(12*k)).
a(2*n) = 2 * a(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(6*sqrt(3)) = 0.949703... (A258414). - Amiram Eldar, Dec 22 2023

A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 16 2007

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

Number 50 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 + q^4 - 2*q^5 - q^8 + 2*q^10 + 2*q^13 + q^16 - 2*q^17 - 2*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A091400, A121363, A121444, A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[2] - A[3] + A[5] - 2*A[6] - A[9] + 2*A[11] + 2*A[14] + A[17] - 2*A[18]; /* Michael Somos, Jul 09 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^3, q^6]^2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] / (2 q^(5/4)), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

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

{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==3, 0, p==2, (-1)^e, p%12>6, !(e%2), (-1)^(e * (p%12==5)) * (e+1))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, -2, -1, -1, 0, 0, -1, -1, -1, 0, 1, -1, -1, -2, -1, -1, 1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, 1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), a(p^e) = e+1 if p == 1 (mod 12), a(p^e) = (-1)^e * (e+1) if p == 5 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(12, k) * x^k/ (1 + x^k + x^(2*k)).
|a(n)| = A091400(n). a(3*n) = a(4*n + 3) = 0. a(2*n) = -a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = - A122856(n). a(4*n + 1) = A121363(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -2 * A121444(n).

A271231 Expansion of the modular cusp form ( eta(q^4) * eta(q^12) )^4 / ( eta(q^2) * eta(q^6) * eta(q^8) * eta(q^24) ), where eta is Dedekind's eta function.

Original entry on oeis.org

0, 1, 0, 1, 0, -2, 0, 0, 0, 1, 0, -4, 0, -2, 0, -2, 0, 2, 0, 4, 0, 0, 0, 8, 0, -1, 0, 1, 0, 6, 0, -8, 0, -4, 0, 0, 0, 6, 0, -2, 0, -6, 0, -4, 0, -2, 0, 0, 0, -7, 0, 2, 0, -2, 0, 8, 0, 4, 0, -4, 0, -2, 0, 0, 0, 4, 0, 4, 0, 8, 0, -8, 0, 10, 0, -1, 0, 0, 0, 8, 0, 1, 0, 4, 0, -4, 0, 6, 0, -6, 0, 0, 0, -8, 0, -8, 0, 2, 0, -4, 0, -18, 0, -16

Offset: 0

Wolfdieter Lang, Apr 19 2016

The modularity pattern of the elliptic curve y^2 = x^3 + x^2 + x considered modulo prime(m) is seen from a(prime(m)) = prime(m) - N(prime(m)) = A271230(m), where N(prime(m))= A271229(m) is the number of solutions of this congruence. That is, the p-defect coincides with the prime indexed expansion coefficient (here for all primes).
This modular cusp form of weight 2 and level N = 48 = 2^4*3 is Nr. 54 in Martin's Table 1 (corrected by giving the 24 the missing exponent -1). See also the Michael Somos link where this correction has been observed.
This modular cusp form is a simultaneous eigenform of every Hecke operators T_p, with p a prime not 2 or 3 (bad primes) with eigenvalue lambda(p) = a(p). (See the Martin reference, Proposition 33, p. 4851.)
In the Martin and Ono reference, p. 3173 (Theorem 2), this cusp form appears (in the corrected version) in the row Conductor 48, and it is there related to the elliptic curve y^2 = x^3 + x^2 - 4*x - 4. The p-defects of this curve coincide with the ones of the curve  y^2 = x^3 + x^2 + x modulo primes p given in A271230. - Wolfdieter Lang, Apr 21 2016
Multiplicative. See A159819 for formula. - Andrew Howroyd, Aug 06 2018

			n=2: a(2) = A271230(1) = 0.
n=5: a(5) = A271230(3) = -2.
See the example section of A271229 for the solutions for the first primes.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A159819, A271229, A271230.

QP = QPochhammer;
a[n_] := If[OddQ[n], SeriesCoefficient[QP[-x] QP[x^2] QP[-x^3] QP[x^6], {x, 0, (n-1)/2}], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 19 2019 *)

q='q+O('q^220); concat([0], Vec( (eta(q^4)*eta(q^12))^4 / (eta(q^2)*eta(q^6)*eta(q^8)*eta(q^24) ) ) ) \\ Joerg Arndt, Sep 12 2016

a(2*n+1) = A159819(n), a(2*n) = 0.
O.g.f.: Expansion in q = exp(2*Pi*i*z) with Im(z) > 0 of (eta(4*z)*eta(12*z))^4 / (eta(2*z)*eta(6*z)*eta(8*z)*eta(24*z)), where eta(z) = q^(1/24)*Product_{n >= 1} (1 - q^n) is the Dedekind function with q = q(z) given above, and i is the imaginary unit.
a(prime(m)) = A271230(m), m >= 1.

A030187 Expansion of eta(q) * eta(q^2) * eta(q^7) * eta(q^14) in powers of q.

Original entry on oeis.org

1, -1, -2, 1, 0, 2, 1, -1, 1, 0, 0, -2, -4, -1, 0, 1, 6, -1, 2, 0, -2, 0, 0, 2, -5, 4, 4, 1, -6, 0, -4, -1, 0, -6, 0, 1, 2, -2, 8, 0, 6, 2, 8, 0, 0, 0, -12, -2, 1, 5, -12, -4, 6, -4, 0, -1, -4, 6, -6, 0, 8, 4, 1, 1, 0, 0, -4, 6, 0, 0, 0, -1, 2, -2, 10, 2, 0

Offset: 1

N. J. A. Sloane

Number 29 of the 74 eta-quotients listed in Table I of Martin (1996).
Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
Coefficients of L-series for elliptic curve "14a4": y^2 + x*y + y = x^3 - x or y^2 + x*y - y = x^3. - Michael Somos, Feb 19 2007

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
LMFDB, Space of modular forms of level 14 and weight 2.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Basis( CuspForms( Gamma1(14), 2), 78)[1]; /* Michael Somos, Nov 20 2014 */

a[ n_] := SeriesCoefficient[ q  QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^7] QPochhammer[ q^14], {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = if( n<1, 0, ellak( ellinit([ -1, 0, -1, -1, 0], 1), n))}; /* Michael Somos, Aug 13 2006 */

{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-1)^e, p==7, 1, a0=1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3 + x^2 - 2*x + 1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^7 + A) * eta(x^14 + A), n))};

CuspForms( Gamma1(14), 2, prec = 78).0; # Michael Somos, Aug 11 2011

Euler transform of period 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -4, ...]. - Michael Somos, Aug 13 2006
a(n) is multiplicative with a(2^e) = (-1)^e, a(7^e) = 1, otherwise a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p . - Michael Somos, Aug 13 2006
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). - Michael Somos, Feb 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = 14 (t / i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(7*k)) * (1 - x^(14*k)).

A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 0, 0, 2, 1, 0, 2, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, -1, 0, -2, 0, 0, -1, 0, 0, 0, 1, -1, -2, 0, 2, 1, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 2, 1, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

sign

Number 52 of the 74 eta-quotients listed in Table I of Martin (1996).

In [Klein and Fricke 1892] on page 586 equation (3) first line left side has A_0 and the right side the power series r^{1/2} (1 - r - r^2 + r^5 + r^7 + ...) which is the g.f. of this sequence. A_0 and the other A_1, A_3, A_9, A_5, A_4 (in a permuted order) correspond to the nonzero 11-sections of the g.f. of this sequence. - Michael Somos, Nov 12 2014

			G.f. = 1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...
G.f. = q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...

F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1892, Vol. 2, see p. 586.
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A006571, A106276.

Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* Michael Somos, Nov 13 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* Michael Somos, Nov 12 2014 *)

{a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 11], n)[n] - qfrep( [3, 1; 1, 4], n)[n])}; /* Michael Somos, Nov 20 2006 */

{a(n) = my(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==11, 1, f = sum( k=0, p-1, (k^3 - k^2 - k - 1)%p == 0); if( f==0, (e-1)%3-1, if( f==1, (1 + (-1)^e) / 2, e+1)))))}; /* Michael Somos, Nov 20 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))}; /* Michael Somos, Nov 20 2006 */

Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Nov 20 2006
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3 - 1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1 + (-1)^e) / 2 if f=1 where f = number of zeros of x^3 - x^2 - x - 1 modulo p. - Michael Somos, Nov 20 2006
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).
a(n) = sum over all solutions to x^2 + x*y + 3*y^2 = 2*n + 1 with odd integer x>0 of (-1)^y. - Michael Somos, Jan 29 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
Convolution square is A006571.

A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

Original entry on oeis.org

1, -3, 0, 2, 9, 0, -22, 0, 0, 26, -6, 0, 25, -27, 0, -46, 0, 0, 26, 66, 0, -22, 0, 0, -45, 0, 0, 0, -78, 0, 74, 18, 0, 122, 0, 0, -46, -75, 0, -142, 81, 0, 0, 0, 0, -44, 138, 0, 2, 0, 0, 194, 0, 0, -214, -78, 0, 0, -198, 0, 121, 0, 0, 146, 66, 0, 52, 0, 0, -22

Offset: 0

N. J. A. Sloane

sign

Number 28 of the 74 eta-quotients listed in Table I of Martin (1996).

Expansion of a newform level 12 weight 3 and character [0, 1].

			G.f. = 1 - 3*x + 2*x^3 + 9*x^4 - 22*x^6 + 26*x^9 - 6*x^10 + 25*x^12 - 27*x^13 + ...
G.f. = q - 3*q^3 + 2*q^7 + 9*q^9 - 22*q^13 + 26*q^19 - 6*q^21 + 25*q^25 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Author?, Eta Products and Quotients which are Newforms. [Broken link?]
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A007332, A152243, A209939.

A := Basis( CuspForms( Gamma1(12), 3), 140); A[1] - 3*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3, {x, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3, n))}; /* Michael Somos, Jun 14 2007 */

Euler transform of period 3 sequence [ -3, -3, -6, ...]. - Michael Somos, Feb 13 2006
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-3)^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^3.
G.f.: Sum_{k>=0} a(k) * q^(2*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * q^(u*u + 3*v*v). - Michael Somos, Jun 14 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008
a(3*n + 1) = -3 * a(n). a(3*n + 2) = 0. a(3*n) = A152243(n). - Michael Somos, Mar 09 2012
a(n) = (-1)^n * A209939(n). - Michael Somos, Mar 16 2012
Convolution square is A007332. - Michael Somos, Nov 16 2008

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A030207 Expansion of eta(q)^2 * eta(q^2) * eta(q^4) * eta(q^8)^2 in powers of q.

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A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

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A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

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A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

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A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

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A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

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