A030203 -id:A030203 - cp's OEIS frontend

A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

1, -9, 27, -9, -117, 216, 27, -450, 459, -9, -648, 1080, -117, -1530, 1350, 216, -1845, 2592, 27, -3258, 2808, -450, -3240, 4752, 459, -5409, 4590, -9, -5850, 7560, -648, -8658, 7371, 1080, -7776, 10800, -117, -12330, 9774, -1530, -11016, 15120, 1350, -16650

Offset: 0

Michael Somos, Jun 17 2005

sign

Number 4 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 313, Equ. (14.2.13).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
George E. Andrews and Bruce C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
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. See p. 697.
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

Cf. A005928, A008654, A103440, A106402, A134340.

A := Basis( ModularForms( Gamma1(3), 3), 44); A[1] - 9*A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, Boole[ n == 0], - 9 DivisorSum[ n, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)

{a(n) = if( n<1, n==0, -9 * sumdiv( n, d, d^2 * kronecker(-3, d)))};

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

G.f.: Product_{k>0} (1 - x^k)^9 / (1 - x^3)^3 = 1 - 9 * Sum_{k>0} x^k * (1 - x^k -6 * x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3.
Expansion of b(q)^3 in powers of q where b() is a cubic AGM theta function.
Euler transform of period 3 sequence [ -9, -9, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u*w * (u + 6*v - 8*w).
Given A = A0 + A1 + A2 is the 3-section, then 0 = A1^3 + A2^3 - 3*A0*A1*A2. A0 = A(q^3) = b(q^3)^3, A1 = -3 * a(q^3)^2 * c(q^3), A2 = 3 * a(q^3) * c(q^3)^2 where a(), b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 19683^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106402. - Michael Somos, Mar 11 2012
a(n) = -9 * A103440(n) unless n = 0. a(6*n + 5) = 216 * A134340(n).
A008654(n) = a(n) + 27 * A106402(n) is the identity a(q)^3 = b(q)^3 + c(q)^3. - Michael Somos, Jul 19 2012
a(n) = -9 * b(n) where b(n) is multiplicative with a(0) = 1, b(p^e) = 1, if p=3, b(p^e) = b(p) * b(p^(e-1)) + Kronecker(-3, p) * p^2 * b(p^(e-2)) otherwise. - Michael Somos, May 18 2015
Convolution cube of A005928. - Michael Somos, May 18 2015

A134177 Expansion of phi(-x^3) * psi(x^4) + x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 11 2007

sign

Number 64 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. = 1 + x - 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7 - 2*x^10 + 3*x^12 + x^13 - ...
G.f. = q + q^3 - 2*q^5 - 2*q^7 + q^9 + 2*q^11 - 2*q^15 - 2*q^21 + 3*q^25 + q^27 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
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. A113780, A128580.

a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] + EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)

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

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

{a(n) = my(A, p, e); 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==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} Kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
a(n) = (-1)^n * A128580(n). a(12*n) = A113780(n).

A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.

Original entry on oeis.org

1, -2, -1, 0, 5, 4, -7, 0, -5, 2, -4, 0, 11, 0, 8, 0, -6, -10, 0, 0, -1, -8, 5, 0, -7, 14, 17, 0, 0, 0, -5, 0, -19, 10, -13, 0, 2, -4, 0, 0, -11, 8, 20, 0, 7, 0, 23, 0, 0, -22, -19, 0, 14, 0, -25, 0, 12, -16, 5, 0, -7, 0, 0, 0, 23, 12, 11, 0, 0, 20, -13, 0, 4, 0, -28, 0, -22, 0, 0, 0, 17, 2, -35, 0, 0, 16, -11, 0, 0, -10

Offset: 0

N. J. A. Sloane

sign

Number 44 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_2(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique weight 2 newform of level 27.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 2*x - x^2 + 5*x^4 + 4*x^5 - 7*x^6 - 5*x^8 + 2*x^9 - 4*x^10 + 11*x^12 + ...
G.f. = q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
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. A030203, A279005.

A := Basis( ModularForms( Gamma0(27), 2), 271); A[2] - 2*A[5]; /* Michael Somos, Jun 12 2014 */

qEigenform( EllipticCurve( [0, 0, 1, 0, 0]), 271); /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(27), 2), 271)[1]; /* Michael Somos, Mar 24 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^2, {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(e/2)), for( i=1, sqrtint(4*p\27), if( issquare(4*p - 27*i^2, &y), break)); a0=1; a1 = y*= (-1)^(y%3); 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<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2, n))}; /* Michael Somos, Feb 19 2007 */

{a(n) = ellak( ellinit( [0, 0, 1, 0, 0], 1), 3*n + 1)}; /* Michael Somos, Jun 12 2014 */

ModularForms( Gamma0(27), 2, prec=271).0; # Michael Somos, Jun 12 2014

Expansion of q^(-1/3) * b(q) * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Coefficients of L-series for elliptic curve "27a3": y^2 + y = x^3. - Michael Somos, Aug 13 2006
Euler transform of period 3 sequence [-2, -2, -4, ...]. - Michael Somos, Dec 06 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2.
a(n) = b(3*n + 1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(e/2), if p == 2 (mod 3), otherwise b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)). - Michael Somos, Aug 13 2006
Given g.f. A(x), then B(q)= q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u*w * (u + 4*w). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
a(4*n + 1) = -2 * a(n). - Michael Somos, Dec 06 2004
a(25*n + 8) = -5 * a(n). Convolution square of A030203. - Michael Somos, Mar 13 2012

A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

Original entry on oeis.org

1, 4, 2, -8, -5, 4, -10, -8, 9, 0, 14, 16, -10, 4, 0, 8, 14, -20, 2, 0, -11, -20, -32, 16, 0, 4, 14, -8, -9, -20, 26, 0, 2, 28, 0, 16, 16, 28, -22, 0, 14, -16, 0, -40, 0, 28, 26, -32, -17, 0, -32, 16, -22, 0, -10, -32, -34, 8, 14, 0, 45, 4, 38, -8, 0, 0, -34

Offset: 0

Michael Somos, Mar 05 2011

sign

Number 67 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. = 1 + 4*x + 2*x^2 - 8*x^3 - 5*x^4 + 4*x^5 - 10*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 4*q^31 - 10*q^37 - 8*q^43 + 9*q^49 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
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. A000727, A187150, A209676, A258779, A280384.

A := Basis( CuspForms( Gamma0(144), 2), 398); A[1] + 4*A[7] + 2*A[11] - 8*A[13]; /* Michael Somos, Jan 01 2017 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, -1], 1), 6*n + 1))};

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

{a(n) = my(m, A, p, e, x, y, a0, a1); if( n<0, 0, m = 6*n + 1; A = factor(m); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-p)^(e/2)), for( y=1, sqrtint(p\3), if( issquare(p - 3*y^2, &x), break)); a0 = 1; if( x%6>3, x = -x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0 = a1; a1 = y); a1)))};

Expansion of f(x)^4 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/6) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^4 in powers of q.
Euler transform of period 4 sequence [4, -8, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = f(t) where q = exp(2 Pi i t).
a(n) = b(6*n + 1) and b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2 * x where p = x^2 + 3 * y^2 == 1 (mod 6) and x == 4, 5 (mod 6).
G.f.: Product_{k>0} (1 - (-x)^k)^4. a(n) = (-1)^n * A000727(n).
Convolution cube is A209676. - Michael Somos, Jun 10 2015
a(2*n) = A258779(n). a(2*n + 1) = 4 * A187150(n). - Michael Somos, Jun 10 2015

A030188 Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.

Original entry on oeis.org

1, -1, -2, 0, 1, 4, -2, 2, 2, -4, 0, -8, -1, -1, 6, 8, -4, 0, 6, 2, -6, 4, -2, 0, -7, -2, -2, -8, 4, 4, -2, 0, 4, -4, 8, 8, 10, 1, 0, -8, 1, -4, -4, -6, -6, 0, -8, 8, 2, 4, -18, 16, 0, -12, -2, -6, 18, 16, -2, 0, 5, 6, 12, -8, -4, -4, 0, 2, -6, -12, 0, -8, -12, 7, 14, -16, 2, -16, -2, 2, 0, 12, 8, 24, -9, -4, 6, 0, -4, 12, 6, 2, -12, 8, 0, 0

Offset: 0

N. J. A. Sloane

sign

Number 43 of the 74 eta-quotients listed in Table I of Martin (1996).

Newform number 1 of degree 1 in Full modular forms space of level 24, weight 2 and trivial character.

			G.f. = 1 - x - 2*x^2 + x^4 + 4*x^5 - 2*x^6 + 2*x^7 + 2*x^8 - 4*x^9 - 8*x^11 + ...
G.f. = q - q^3 - 2*q^5 + q^9 + 4*q^11 - 2*q^13 + 2*q^15 + 2*q^17 - 4*q^19 + ...

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.3.

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. A030209, A159819. A258090.

Basis( CuspForms( Gamma0(24), 2), 192) [1]; /* Michael Somos, May 27 2014 */

qEigenform( EllipticCurve( [0, -1, 0, 1, 0]), 191); /* Michael Somos, Jun 12 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6], {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^2 + A) * eta(x^3 + A) * eta(x^6 + A), n))}; /* Michael Somos, Apr 02 2005 */

{a(n) = if( n<0, 0, n = 2*n + 1; ellak( ellinit([ 0, -1, 0, -4, 4], 1), n))}; /* Michael Somos, Apr 02 2005 */

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

CuspForms( Gamma0(24), 2, prec=192).0 # Michael Somos, May 24 2013

Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. - Michael Somos, Apr 02 2005
Given g.f. A(x), then B(x) = x * A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = u^2 * v * w + 4 * u * v^2 * w + 16 * u * v * w^2 + 4 * u^2 * w^2 - v^4. - Michael Somos, Apr 02 2005
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise where b(p) = p+1 - number of solutions to y^2 = x^3 - x^2 - 4*x + 4 modulo p including the point at infinity. - Michael Somos, Mar 04 2011
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
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). - Michael Somos, Jun 08 2007
Coefficients of L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - Michael Somos, Apr 02 2005
a(n) = (-1)^n * A159819(n). a(3*n + 1) = -a(n). Convolution square is A030209. - Michael Somos, Mar 13 2012
a(3*n + 2) = -2 * A258090(n). - Michael Somos, May 19 2015

A030204 Expansion of q^(-1/8) * eta(q) * eta(q^2) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Number 66 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).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

			G.f. = 1 - x - 2*x^2 + x^3 + 2*x^5 + x^6 - 2*x^9 + x^10 - 2*x^11 - 2*x^12 + ...
G.f. = q - q^9 - 2*q^17 + q^25 + 2*q^41 + q^49 - 2*q^73 + q^81 - 2*q^89 - 2*q^97 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
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
Michael Somos, Introduction to Ramanujan theta functions
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334. [a(n) = coefficient of q^(9m+1) in the q-expansion of the unique normalized newform g of weight 1, level 128, and character chi_2. - _N. J. A. Sloane_, Oct 18 2014]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A002513, A107063, A107064, A138514.

Basis( CuspForms( Gamma1(128), 1), 641)[1]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, I x] / (4 Sqrt[ x] I^(1/4)), {x, 0, 4 n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / (2^(3/2) x^(1/2)), {x, 0, 4 n}]; (* Michael Somos, Jan 31 2015 *)

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

{a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e + 1) * if( qfbclassno(-4*p)%8, (-1)^e, 1), e%2==0, (-1)^(e/2*(p%8<5)))))}; /* Michael Somos, Jul 26 2006 */

{a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0;0, 32], n) - qfrep([4, 2; 2, 9], n))[n])}; /* Michael Somos, Sep 02 2006 */

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)).
G.f.: (Sum_{k>0} x^((k^2 - k)/2)) * (Sum_{k in Z} (-1)^k * x^k^2). - Michael Somos, Sep 02 2006
Expansion of psi(x) * phi(-x) = f(-x^2) * f(-x) = f(-x)^2 / chi(-x) = f(-x)^3 / phi(-x) = f(-x^2)^2 * chi(-x) = f(-x^2)^3 / psi(x) = psi(-x) * phi(-x^2) = psi(x)^2 * chi(-x)^3 = phi(-x)^2 / chi(-x)^3 = (f(-x)^3 * psi(x))^(1/2) = (f(-x^2)^3 * phi(-x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2008
Expansion of psi(x) * psi(-x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Oct 11 2013
Euler transform of period 2 sequence [ -1, -2, ...].
a(3*n) = A107063(n). a(3*n + 2) = -2 * A107064(n). - Michael Somos, Oct 11 2013
a(9*n + 1) = -a(n), a(9*n + 4) = a(9*n + 7) = 0. - Michael Somos, Mar 17 2004
a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p === 3,5,7 (mod 8) and e odd, b(p^e) = (-1)^(e/2) if p == 3 (mod 8) and e even, b(p^e) = 1 if p == 5,7 (mod 8) and e even, b(p^e) = e + 1 if p == 1 (mod 8) and p = x^2 + 32*y^2, b(p^e) = (-1)^e * (e + 1) if p == 1 (mod 8) and p is not of the form x^2 + 32*y^2.
a(n) = (-1)^n * A138514(n). Convolution inverse is A002513.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

Original entry on oeis.org

1, -2, -1, 2, 1, 0, 2, 2, -6, -4, -4, 6, 1, 4, 6, -4, 0, -2, 2, -4, 6, -10, -1, -6, -3, 12, -6, 0, 8, 12, 2, 2, -2, 2, -12, -12, 2, -2, 0, 8, -11, 6, 6, -12, -6, 4, 8, 4, 2, 0, 6, 14, 4, -6, 2, -4, -6, -6, 2, -12, -11, -12, -1, 2, 20, 0, -8, -4, 18, -4, 12, 0

Offset: 0

N. J. A. Sloane

sign

This eta-quotient of conductor 20 is one of the twelve weight 2 newforms listed by Martin and Ono.
The associated elliptic curve is "20a1": y^2 = x^3 + x^2 + 4*x + 4 or "20a2": y^2 = x^3 + x^2 - x.
Number 39 of the 74 eta-quotients listed in Table I of Martin (1996).
The mentioned eta-quotient is in fact eta^2(2*z) * eta^2(10*z) with q = exp(2*Pi*i*tau) with Im(tau) > 0, I^2 = -1, with the q-expansion coefficients b(n) from the Michael Somos Oct Aug 13 2006 formula: b(2*n) = 0 and b(2*n+1) = a(n), for n >= 0. A273163(k) = b(prime(k)), k >= 1. See also the comments on multiplicativity of b(n) (called there c(n)) with b(2^k) = b(2)^k = 0, b(5^k) = b(5)^k = (-1)^k, and b(prime(n)^k) = (sqrt(prime(n)))^k*S(k,A273163(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. Compare this with the b(p^(e+2)) recurrence given by Michael Somos, Oct 31 2005. - Wolfdieter Lang, May 23 2016

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^6 + 2*x^7 - 6*x^8 - 4*x^9 - 4*x^10 + ...
G.f. of b(n) from eta^2(2*z)*eta^2(10*z) = q - 2*q^3 - q^5 + 2*q^7 + q^9 + 2*q^13 + 2*q^15 - 6*q^17 - 4*q^19 + ..., where q = exp(2*Pi*I*z).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 20 and weight 2.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
K. Ono, Ramanujan, taxicabs, birthdates, ZIP codes and twists, Amer. Math. Monthly, 104 (1997), 912-917. MR1490909 (98i:11020)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030210, A049310, A159817, A273163.

Basis( CuspForms( Gamma0(20), 2), 145) [1]; /* Michael Somos, May 27 2014 */

A := Basis( CuspForms( Gamma1(20), 2), 145); A[1] - 2*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, 4, 4], 1), n)[n])}; /* Michael Somos, Oct 31 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^2, n))}; /* Michael Somos, Oct 31 2005 */

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, -1, 0], 1), n)[n])}; /* Michael Somos, Aug 13 2006 */

{a(n) = my(A, p, e, x, y, a0, a1); 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==5, (-1)^e, a0=1; a1 = y = -sum( x=0, p-1, kronecker( x^3 + x^2 - x, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma0(20), 2, prec=92).0; # Michael Somos, May 28 2013

Euler transform of period 5 sequence [ -2, -2, -2, -2, -4, ...]. - Michael Somos, Oct 31 2005
Given g.f. A(x), then B(x) = q * A(q)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u*w * (u + 8*v + 16*w) - v^3. - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, else b(p^(e+2)) = b(p)*b(p^(e+1)) - p*b(p^e). - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) and b(p) = p minus number of points of elliptic curve "20a1" or "20a2" modulo p. - Michael Somos, Aug 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^2.
a(121*n + 60) = -11 * a(n).
Convolution square is A030210. - Michael Somos, Jun 13 2014
a(n) = (-1)^n * A159817(n). - Michael Somos, Jun 10 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A030211 Expansion of q^(-1/2) * (eta(q) * eta(q^2))^4 in powers of q.

Original entry on oeis.org

1, -4, -2, 24, -11, -44, 22, 8, 50, 44, -96, -56, -121, 152, 198, -160, 176, -48, -162, -88, -198, 52, 22, 528, 233, -200, -242, 88, -176, -668, 550, -264, -44, 188, 224, 728, 154, 484, -1056, -656, -311, 236, -100, -792, 714, 528, 640, -88, -478, 484, 1566, -968, 192, -780, -1994, 648, -942

Offset: 0

N. J. A. Sloane

This is Glaisher's P(n). - N. J. A. Sloane, Nov 24 2018

Number 16 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x - 2*x^2 + 24*x^3 - 11*x^4 - 44*x^5 + 22*x^6 + 8*x^7 + 50*x^8 + ...
G.f. = q - 4*q^3 - 2*q^5 + 24*q^7 - 11*q^9 - 44*q^11 + 22*q^13 + 8*q^15 + ...

J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See p. 340.
J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m). Quart. J. Math, 37 (1906), 36-48.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
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
Eric Weisstein's World of Mathematics, Apéry Number.
Index entries for sequences mentioned by Glaisher

Cf. A002171, A134461 (the same except for signs).

Basis( CuspForms( Gamma0(8), 4), 115) [1]; /* Michael Somos, May 27 2014 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^4, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = if( n<0, 0, polcoeff( (eta(x + x * O(x^n)) * eta(x^2 + x * O(x^n)))^4, n))}; /* Michael Somos, Apr 14 2004 */

q='q+O('q^99); Vec((eta(q)*eta(q^2))^4) \\ Altug Alkan, Sep 19 2018

CuspForms( Gamma0(8), 4, prec=115).0; # Michael Somos, May 28 2013

G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^4.
Euler transform of period 2 sequence [ -4, -8, ...]. - Michael Somos, Apr 14 2004
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (81*u6*u3 + u1*u2) * (u2*u3 + u1*u6) + 30 * u1*u2*u3*u6 - 256 * u2^2*u6^2 - 5 * u2^2*u3^2 - 5 * u1^2*u6^2 - u1^2*u3^2. - Michael Somos, Mar 08 2006
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = 8 * A0*A2 * (A0^2 + A2^2) + (A1^2 - A3^2) * (A0^2 - A2^2). - Michael Somos, Mar 08 2006
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013
a(n) = (-1)^n * A134461(n). Convolution square of A002171.
G.f.: exp(4*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A030212 Glaisher's chi_4(n).

Original entry on oeis.org

1, -4, 0, 16, -14, 0, 0, -64, 81, 56, 0, 0, -238, 0, 0, 256, 322, -324, 0, -224, 0, 0, 0, 0, -429, 952, 0, 0, 82, 0, 0, -1024, 0, -1288, 0, 1296, 2162, 0, 0, 896, -3038, 0, 0, 0, -1134, 0, 0, 0, 2401, 1716, 0, -3808, 2482, 0, 0, 0, 0, -328, 0, 0, -6958, 0, 0, 4096, 3332, 0, 0, 5152, 0, 0

Offset: 1

N. J. A. Sloane

Number 10 of the 74 eta-quotients listed in Table I of Martin (1996). Cusp form level 4 weight 5.
Called chi_4(n) by Glaisher and Hardy because as Glaisher (1907) writes on page 21 "It can be shown (see section 53) that chi_4(n) admits of an arithmetical definition, being in fact equal to one-fourth of the sum of the fourth powers of all complex numbers which have n as norm, viz. chi_4(n) = 1/4 sum_n (a + i b)^4, where a + i b is any number which has n for norm. It is in consequence of this definition that the notation chi_4(n) has been used." - Michael Somos, Jun 18 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 4*q^2 + 16*q^4 - 14*q^5 - 64*q^8 + 81*q^9 + 56*q^10 - 238*q^13 + ...
From _Seiichi Manyama_, Apr 25 2017: (Start)
a(1) = (1 + 0i)^4 = 1,
a(2) = (1 + 1i)^4 = -4,
a(4) = (2 + 0i)^4 = 16,
a(5) = (1 + 2i)^4 + (2 + 1i)^4 = -7 - 24i - 7 + 24i = -14,
a(8) = (2 + 2i)^4 = -64,
a(9) = (3 + 0i)^4 = 81,
a(10) = (1 + 3i)^4 + (3 + 1i)^4 = 28 - 96i + 28 + 96i = 56 (End)

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 175, 4.7 Exercise 5. MR1471703 (98g:14032)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 34).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Newform orbit 4.5.b.a.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=10, called F(tau).
M. Rogers, J.G. Wan, and I. J. Zucker, Moments of Elliptic Integrals and Critical L-Values, arXiv:1303.2259 [math.NT], 2013.
Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?
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
Index entries for sequences mentioned by Glaisher

Cf. A002607, A215472, A247067.

Basis( CuspForms( Gamma1(4), 5), 71) [1]; /* Michael Somos, May 27 2014 */

If[SquaresR[2,#]==0,0,1/4 Plus@@((x+I y)^4/.{ToRules[Reduce[x^2+y^2==#,{x,y},Integers]]})] &/@Range[70] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^4 * eta(x^2 + A)^2, n))}; /* Michael Somos, Jul 17 2004 */

{a(n) = local(r); if( n<1, 0, r = sqrtint(n); sum( x=-r, r, sum( y=-r, r, if( x^2 + y^2 == n, (x + I*y)^4) )) / 4 )}; /* Michael Somos, Sep 12 2005 */

{a(n) = my(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, (-4)^e, p%4 == 3, if( e%2, 0, p^(2*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^4 ) * 2; for( i=2, e, y = x*a1 - p^4*a0; a0=a1; a1=y); a1))) }; /* Michael Somos, Nov 18 2014 */

CuspForms( Gamma1(4), 5, prec=71).0; # Michael Somos, May 28 2013

Expansion of phi(q)^2 * psi(-q)^8 = chi(q)^6 * psi(-q)^10 = f(q)^3 * psi(-q)^7 = f(-q^2)^6 * psi(-q)^4 = f(-q^2)^10 / chi(q)^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 12 2013
Expansion of eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 in powers of q.
G.f.: x * (Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4).
G.f.: (t*t'' - 3(t')^2) / 2 where t = theta_3(x) (A000122) and t' := x * (dt/dx), t'' := (t')'. - Michael Somos, Nov 08 2005
Euler transform of period 4 sequence [ -4, -6, -4, -10, ...]. - Michael Somos, Jul 17 2004
a(n) is multiplicative with a(2^e) = (-4)^e, a(p^e) = p^(2*e) * (1 + (-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p) * a(p^(e-1)) - p^4 * a(p^(e-2)) for p == 1 (mod 4) where a(p) = 2 * Re( (x + i*y)^4 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = (A0^2 - A2^2)^2 + 4 * A0*A2*A1^2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013
a(4*n + 3) = 0. - Michael Somos, Mar 12 2013
a(2*n) = -4 * a(n). a(4*n + 1) = A215472(n). - Michael Somos, Sep 05 2013
a(n) = 1/4 * Sum_{a^2 + b^2 = n} (a + bi)^4 = Sum_{a > 0, b >= 0, a^2 + b^2 = n} (a + bi)^4. - Seiichi Manyama, Apr 25 2017

A030184 Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) in powers of q.

Original entry on oeis.org

1, -1, -1, -1, 1, 1, 0, 3, 1, -1, -4, 1, -2, 0, -1, -1, 2, -1, 4, -1, 0, 4, 0, -3, 1, 2, -1, 0, -2, 1, 0, -5, 4, -2, 0, -1, -10, -4, 2, 3, 10, 0, 4, 4, 1, 0, 8, 1, -7, -1, -2, 2, -10, 1, -4, 0, -4, 2, -4, 1, -2, 0, 0, 7, -2, -4, 12, -2, 0, 0, -8, 3, 10, 10, -1

Offset: 1

N. J. A. Sloane

Unique cusp form of weight 2 for congruence group Gamma_1(15). - Michael Somos, Aug 11 2011
Coefficients of L-series for elliptic curve "15a8": y^2 + x*y + y = x^3 + x^2 or y^2 + x*y - y = x^3 + x^2 + x. - Michael Somos, Feb 01 2004
Number 32 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 - q^10 - 4*q^11 + q^12 - 2*q^13 - ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.
LMFDB, Space of modular forms of level 15 and weight 2.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Basis( CuspForms( Gamma1(15), 2), 76)[1]; /* Michael Somos, Nov 20 2014 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ q^15], {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = if( n<1, 0, ellak( ellinit([ 1, 1, 1, 0, 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==3, (-1)^e, p==5, 1, a0=1; a1 = y = -if( p==2, 1, sum(x=0, p-1, kronecker( 4*x^3 + 5*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^3 + A) * eta(x^5 + A) * eta(x^15 + A), n))}; /* Michael Somos, May 02 2005 */

CuspForms( Gamma1(15), 2, prec = 76).0; # Michael Somos, Aug 11 2011

Euler transform of period 15 sequence [ -1, -1, -2, -1, -2, -2, -1, -1, -2, -2, -1, -2, -1, -1, -4, ...]. - Michael Somos, May 02 2005
a(n) is multiplicative with a(5^e) = 1, a(3^e) = (-1)^e, 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^3 - u*w* (u + 2*v + 4*w). - Michael Somos, May 02 2005
G.f. A(x) satisfies 2 * A(x^2) = -(A(x) + A(-x) + 4*A(x^4)), A(x^2)^3 = -A(x) * A(-x) * A(x^4). - Michael Somos, Feb 19 2007
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(15*k)).

cp's OEIS Frontend

A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A134177 Expansion of phi(-x^3) * psi(x^4) + x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Magma

Mathematica

PARI

PARI

PARI

Sage

Formula

A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A030188 Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

PARI

PARI

Sage