A030203 -id:A030203 - cp's OEIS frontend

A030210 Expansion of (eta(q) * eta(q^5))^4 in powers of q.

1, -4, 2, 8, -5, -8, 6, 0, -23, 20, 32, 16, -38, -24, -10, -64, 26, 92, 100, -40, 12, -128, -78, 0, 25, 152, -100, 48, -50, 40, -108, 256, 64, -104, -30, -184, 266, -400, -76, 0, 22, -48, 442, 256, 115, 312, -514, -128, -307, -100, 52, -304, 2, 400, -160, 0, 200, 200, 500, -80, -518, 432, -138, -512

Offset: 1

N. J. A. Sloane

Conjecture: |a(p)| < 2*p^(3/2) for p prime. - Michael Somos, Oct 31 2005
Unique cusp form of weight 4 for congruence group Gamma_1(5). - Michael Somos, Aug 11 2011
Number 13 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 4*q^2 + 2*q^3 + 8*q^4 - 5*q^5 - 8*q^6 + 6*q^7 - 23*q^9 + 20*q^10 + 32*q^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006 , pp. 107-115. See function c5 pp. 107-108.
LMFDB, Space of modular forms of level 5 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. A030205.

Basis( CuspForms( Gamma1(5), 4), 65) [1]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^5])^4, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = my(A); if( n<1, 0, n--; A = x^n * O(x); polcoeff( (eta(x + A) * eta(x^5 + A))^4, n))}

CuspForms( Gamma1(5), 4, prec = 65).0 # Michael Somos, Aug 11 2011

Euler transform of period 5 sequence [ -4, -4, -4, -4, -8, ...]. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 8*u*v*w + 16*u*w^2 + u^2*w. - Michael Somos, May 02 2005
a(n) is multiplicative with a(5^e) = (-5)^e, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (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^(5*k)))^4.
Convolution square of A030205. - Michael Somos, Jun 15 2014

A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

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

Offset: 0

Michael Somos, Apr 22 2009

sign

Number 54 of the 74 eta-quotients listed in Table I of Martin (1996).
Table I of Martin (1996) for this q-series has exponent of 24 wrong. Number 54 should read 2^(-1)*4^4*6^(-1)*8^(-1)*12^4*24^(-1) (in column g).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The present expansion corresponds in Martin's notation to
  1^(-1)*2^4*3^(-1)*4^(-1)*6^4*12^(-1). For the expansion of the (corrected) Nr. 54 of Martin's reference see A271231. One finds for the p-defects prime(m) - N(prime(m)) = A271230(m) of the elliptic curve y^2 = x^3 + x^2 + x (mod prime(m)), where N(prime(n)) = A271229(n) is the number of solutions, the modularity pattern A271231(prime(m)) = A271230(m), m >= 1. - Wolfdieter Lang, Apr 18 2016

			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 + ...

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.
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. A030188, A271229, A271230, A271231.

A := Basis( CuspForms( Gamma0(48), 2), 147); A[1] + A[3]; /* Michael Somos, Mar 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Mar 31 2015 *)

{a(n) = if(n<0, 0, ellak( ellinit([0, 1, 0, 1, 0], 1), 2*n + 1))};

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

{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, 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)))};

q='q+O('q^220); Vec( (eta(q^2)*eta(q^6))^4 / (eta(q^1)*eta(q^3)*eta(q^4)*eta(q^12) ) ) \\ Joerg Arndt, Sep 12 2016

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Expansion of f(x) * f(-x^2) * f(x^3) * f(-x^6) in powers of x where f() is a Ramanujan theta function.
Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -6, 1, -2, 2, -3, 1, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - (-x)^k) * (1 - x^(2*k)) * (1 - (-x)^(3*k)) * (1 - x^(6*k)).
a(n) = (-1)^n * A030188(n).

A271230 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + x (mod p) for primes p, where N(p) is the number of solutions.

Original entry on oeis.org

0, 1, -2, 0, -4, -2, 2, 4, 8, 6, -8, 6, -6, -4, 0, -2, -4, -2, 4, -8, 10, 8, 4, -6, 2, -18, -16, 12, -2, 18, 8, 4, -6, 12, 14, 16, -2, -12, -24, 6, -12, 6, 0, 2, -18, -16, 20, 8, -12, 22, 10, 16, 18, -20, 2, 8, -10, -8, -26, 26

Offset: 1

Wolfdieter Lang, Apr 18 2016

The modularity pattern series is the expansion of the (corrected) Nr. 54 modular cusp form of weight 2 and level N=48 given in the table 1 of the Martin reference, i.e., (eta(4*z) * eta(12*z)^4 / (eta(2*z) * eta(6*z) * eta(8*z) * eta(24*z)) in powers of q = exp(2*Pi*i*z), with Im(z) > 0, where i is the imaginary unit. Here eta(z) = q^{1/24}*Product_{n>=1} (1-q^n) is the Dedekind eta function. See A271231 for this expansion. Note that also for the possibly bad prime 2 and the bad prime 3 (the discriminant of this elliptic curve is -3) this expansion gives the correct p-defect.

The identical p-defects occur for the elliptic curve y^2 = x^3 + x^2 - 4*x - 4 taken modulo prime(n). See the Martin and Ono reference, p. 3173, row Conductor 48, and A271231 (checked up to prime(100) = 541). - Wolfdieter Lang, Apr 21 2016

			See the example section of A271229.
n = 3, prime(3) = 5, A271229(5) = 7, a(3) = 5 - 7 = -2.

Seiichi Manyama, Table of n, a(n) for n = 1..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
Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong and Jinquan Luo, The Differential Spectrum of the Power Mapping x^(p^n-3), arXiv:2108.03088 [cs.IT], 2021. See Table II p. 7.

Cf. A159819, A271229, A271231.

a(n) = prime(n) - A271229(n), n >= 1, where A271229(n) is the number of solutions of the congruence y^2 == x^3 + x^2 + x (mod prime(n)).

a(n) = A271231(prime(n)), n >=1.

A002655 Expansion of Product_{i >= 1} (1 - q^i)(1 - q^(7i)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Number 56 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - x - x^2 + x^5 + x^8 + x^9 - 2*x^12 - 2*x^14 + x^16 - x^21 + 2*x^22 + ...
G.f. = q - q^4 - q^7 + q^16 + q^25 + q^28 - 2*q^37 - 2*q^43 + q^49 - q^64 + ...

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.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A160806.

Basis( CuspForms( Gamma1(63), 1), 242) [1]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^7], {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

{a(n) = if( n<0, 0, n = 3*n + 1; (qfrep( [2, 1; 1, 32], n, 1) - qfrep( [8, 1; 1, 8], n, 1))[n])}; /* Michael Somos, May 28 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A), n))}; /* Michael Somos, May 28 2005 */

{a(n) = my(A, p, e, x, y); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, real(I^e), p==3, 0^e, p==7, (-1)^e, kronecker( p, 7)==-1, !(e%2), for( x=0, sqrtint(p\7), if( issquare(p - 7*x^2, &y), y = if( x%3 && y%3, real(I^e), (e+1) * if( x%3, (-1)^e,1)); break)); y)))}; /* Michael Somos, May 28 2005 */

Expansion of q^(-1/3) * eta(q) * eta(q^7) in powers of q.
Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w + 2 * u*w^2 - v^3 - Michael Somos, Dec 06 2004
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(7*k)).
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^(2*e)) = (-1)^e if p=2, b(p^e) = 0^e if p = 3, b(p^e) = (-1)^e if p = 7, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), else p == 1, 2, 4 (mod 7) and p = y^2 + 7*x^2 when b(p^(2*e)) = (-1)^e if x*y not divisible by 3, b(p^e) = e+1 if x divisible by 3 or (e+1) * (-1)^e if y divisible by 3 . - Michael Somos, May 28 2005
a(2*n) = A160806(n). a(4*n + 3) = 0. a(4*n + 1) = -a(n). a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) =  63^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 17 2015

A002656 Expansion of (eta(q) * eta(q^7))^3 in powers of q.

Original entry on oeis.org

1, -3, 0, 5, 0, 0, -7, -3, 9, 0, -6, 0, 0, 21, 0, -11, 0, -27, 0, 0, 0, 18, 18, 0, 25, 0, 0, -35, -54, 0, 0, 45, 0, 0, 0, 45, -38, 0, 0, 0, 0, 0, 58, -30, 0, -54, 0, 0, 49, -75, 0, 0, -6, 0, 0, 21, 0, 162, 0, 0, 0, 0, -63, -91, 0, 0, -118, 0, 0, 0, 114, -27

Offset: 1

N. J. A. Sloane

Number 15 of the 74 eta-quotients listed in Table I of Martin (1996).

Unique cusp form of weight 3 for congruence group Gamma_1(7). - Michael Somos, Aug 11 2011

			G.f. = q - 3*q^2 + 5*q^4 - 7*q^7 - 3*q^8 + 9*q^9 - 6*q^11 + 21*q^14 - 11*q^16 + ...

B. Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 372 (4).
N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 145, problem 13.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. Elkies, The Klein quartic in number theory.
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), no. 1, 1-17. see pp 9-10.
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

Basis( CuspForms( Gamma1(7), 3), 72) [1]; /* Michael Somos, Dec 09 2013 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^7])^3, {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^7 + A))^3, n))}; /* Michael Somos, Apr 16 2005 */

{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( kronecker(-7, p)<1, if( p==7, (-1)^e, 1-e%2) * p^e, for(i=1, sqrtint(p\7), if( issquare(p - 7*i^2), y=i; break)); a0 = 1; a1 = y = if( p==2, -3, 2*p - 28*y^2); for(i=2, e, x = y * a1 - p^2 * a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Oct 19 2005 */

CuspForms( Gamma1(7), 3, prec = 72).0; # Michael Somos, Aug 11 2011

Euler transform of period 7 sequence [ -3, -3, -3, -3, -3, -3, -6, ...]. - Michael Somos, Mar 11 2004
a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), a(p^e) = a(p) * a(p^(e-1)) - p^2 * a(p^(e-2)) and a(2) = -3, a(p) = 2 * (x^2 - 7*y^2) where p = x^2 + 7*y^2 if p == 1, 2, 4 (mod 7). - Michael Somos, Apr 12 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u * w * (u + 6*v + 8*w) - v^3. - Michael Somos, May 02 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 12 2008
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(7*k)))^3. - Michael Somos, Aug 11 2011
G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + u*v + 2*v*v). - Michael Somos, Jun 14 2007
a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. - Michael Somos, Oct 19 2005

A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96

Offset: 0

Michael Somos, Oct 29 2005

sign

Number 46 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 + 2*x - 4*x^2 - 8*x^3 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...

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. A109039, A113417, A209940, A258096.

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)

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

{a(n) = my(A, p, e, t); 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, t = (-1)^(p\2); p *= kronecker( -2, p); (p^(e+1) - t^(e+1)) / (p - t) )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};

Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - Michael Somos, Mar 14 2012
Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).
G.f.: Sum_{k>0} (2*k - 1) * (-1)^[(k - 1)/2] * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - Michael Somos, May 20 2015

A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

Original entry on oeis.org

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604

Offset: 0

Michael Somos, Jun 05 2006

sign

Number 8 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*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

G. C. Greubel, 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. 37.
Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
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. A000141, A002173, A050470, A128692.

A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)

{a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};

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

Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
a(n) = -4 * A002173(n) unless n=0.
Convolution of A000141 and A128692.

A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 0, 4, 1, 2, 0, 3, 0, 0, 1, 5, 2, 2, 2, 3, 0, 0, 2, 4, 1, 0, 1, 0, 0, 2, 2, 6, 0, 4, 0, 3, 0, 4, 0, 4, 0, 0, 0, 0, 1, 4, 2, 5, 1, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2, 4, 0, 7, 0, 0, 0, 6, 2, 0, 0, 4, 0, 0, 1, 6, 0, 0, 2, 5, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 4, 2, 6, 0, 2, 0, 3, 0, 4, 0, 0

Offset: 0

Michael Somos, Oct 14 2006

Number 31 of the 74 eta-quotients listed in Table I of Martin (1996).

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence Kronecker(-15, n). - Andrew Howroyd, Jul 27 2018

			G.f. = 1 + q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..1000
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.

Cf. A035175, A106406.

A := Basis( ModularForms( Gamma1(15), 1), 106); A[1] + A[2] + 2*A[3] + A[4] + 3*A[5] + A[6] + 2*A[7]; /* Michael Somos, Feb 10 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^5])^2 / ( QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Feb 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ KroneckerSymbol[ -15, d], { d, Divisors[ n]}]]; (* Michael Somos, Feb 10 2015 *)

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d)))};

{a(n) = if( n<1, n==0, (qfrep( [2, 1; 1, 8],n, 1) + qfrep( [4, 1; 1, 4], n, 1))[n])};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...].
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = - v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(n).
a(n) = A035175(n) unless n=0. a(n) = |A106406(n)| unless n=0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 10 2015
a(n) = Sum_{d | n}  Kronecker(-15, d). - Andrew Howroyd, Jul 27 2018
From Amiram Eldar, Feb 20 2024: (Start)
Multiplicative with a(p^e) = 1 if p = 3 or 5, e + 1 if Kronecker(-15, p) = 1, and 1 - (e mod 2) if Kronecker(-15, p) = -1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15). (End)

A138514 Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) 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, -2, 0, 2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, -2, 0, 0, -2

Offset: 0

Michael Somos, Mar 22 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 70 of the 74 eta-quotients listed in Table I of Martin (1996).
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 + ...

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.
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. A030204, A083650.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + 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( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))) };

{a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([ 1, 0; 0, 64], n) - qfrep([ 4, 2; 2, 17], n))[n])};

Expansion of f(x) * f(-x^2) = psi(-x) * phi(x) = chi(x) * f(-x^2)^2 = psi(x) * phi(-x^2) = f(x)^2 / chi(x) = f(x)^3 / phi(x) = f(-x^2)^3 / psi(-x) = phi(x)^2 / chi(x)^3 = chi(x)^3 * psi(-x)^2 = (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.
Expansion of psi(i * x) * psi(-i * x) in powers x^2 where i^2 = -1 and psi() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].
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 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.
a(9*n + 1) = a(n), a(9*n + 4) = a(9*n + 7) = 0. a(n) = (-1)^n * A030204(n) = (-1)^floor((n+1)/2) * A083650(n).
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 16 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.

Original entry on oeis.org

1, 2, -3, -6, 2, 0, -1, 10, 0, 2, 10, -6, -7, -14, 0, 10, -12, 0, -6, 0, 9, 4, 10, 0, 18, 2, 0, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 20, 0, 0, 0, -12, 3, -30, 26, 0, -30, -14, 0, 0, 2, -30, -28, 26, 0, 18, 10, 0, -13, 34, 0, 0, 20, 0, 26, -22, 0, 6, 0, -6, 18, 0

Offset: 0

Michael Somos, Mar 22 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 16 2012
The weight 2 eta-quotient newform  eta^8(8*z) / (eta^2(4*z)*eta^2(16*z)) appears in Theorem 2 of the Martin and Ono link in the row with conductor 64 for the strong Weil curve y^2 = x^3 - 4*x. For N(p), the number of solutions modulo primes for this elliptic curve and for y^2 = x^3 + x, see A095978. The non-vanishing p-defects p - N(p) for these two curves are given in A267859.  - Wolfdieter Lang, May 26 2016

			G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 - x^6 + 10*x^7 + 2*x^9 + 10*x^10 - 6*x^11 + ...
G.f. for {b(n)} = q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 - 6*q^45 - 7*q^49 + ...

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)
LMFDB, Elliptic Curve 64.a4 (Cremona label 64a4)
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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A095978, A138514, A267859, A280339.

A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* Michael Somos, May 15 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)

{a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)};

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

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

Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -6, 2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) where p = x^2 + 4 * y^2.
G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2.
a(n) = (-1)^n * A002171(n). a(9*n + 2) = -3 * a(n), a(9*n + 5) = a(9*n + 8) = 0. Convolution square of A138514.
G.f. for{b(n)}:
  eta^8(8*z)/(eta^2(4*z)*eta^2(16*z)) with q = exp(2*Pi*i*z)), Im(z) > 0 (see a comment on the Martin-Ono link above). - Wolfdieter Lang, May 27 2016

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A030210 Expansion of (eta(q) * eta(q^5))^4 in powers of q.

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A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

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A271230 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + x (mod p) for primes p, where N(p) is the number of solutions.

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A002655 Expansion of Product_{i >= 1} (1 - q^i)*(1 - q^(7*i)).

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A002656 Expansion of (eta(q) * eta(q^7))^3 in powers of q.

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A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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A002655 Expansion of Product_{i >= 1} (1 - q^i)(1 - q^(7i)).