A030205 -id:A030205 - 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

A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 5, 6, 5, 11, 11, 23, 23, 17, 23, 35, 35, 35, 53, 53, 59, 47, 59, 65, 83, 71, 71, 77, 95, 95, 95, 89, 113, 107, 119, 125, 131, 119, 143, 155, 131, 179, 173, 149, 179, 191, 191, 203, 167, 179, 191, 227, 233, 233, 215, 239, 263, 227, 251, 263, 281, 251, 251, 251, 275

Offset: 1

Wolfdieter Lang, May 20 2016

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the fourth row, starting with conductor 20, as a strong Weil curve for the weight 2 newform (eta(2*z)*eta(10*z))^2, symbolically 2^2 10^2, with Im(z) > 0, and the Dedekind eta function. See A030205 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z)*(eta(z)*eta(5*z))^2. For the q-expansion of (eta(2*z)*eta(10*z))^2 one has interspersed 0's: 0, 1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, ... This modular cusp form of weight 2 appears as the 39th entry in Martin's Table I.
For the p-defect prime(n) - a(n) see A273163(n), n >= 1.
The discriminant of this elliptic curve is -400 = -2^4*5^2 (bad primes 2 and 5, also the prime divisors of the conductor).
The congruence y^2 == x^3 + x^2 - x has the same number of solutions modulo prime(n). See a comment on A030205. The discriminant equals +5.

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 0)
2,   3,       5:   (0, 1), (0, 2), (1, 1),
                   (1, 2) (2, 0)
3,   5,       6:   (0, 2), (0, 3), (1, 0),
                   (2, 2), (2, 3), (4, 0)
4,   7,       5:   (0, 2), (0, 5), (4, 3),
                   (4, 4), (6, 0)
5,  11,      11:   (0, 2), (0, 9), (4, 1),
                   (4, 10), (5, 3), (5, 8),
                   (6, 4), (6, 7), (9, 5),
                   (9, 6), (10, 0)
...
The solutions (x, y) of y^2 == x^3 + x^2 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       5:   (0, 0), (1, 1), (1, 2),
                   (2, 1) (2, 2)
3,   5,       6:   (0, 0), (1, 1), (1, 4),
                   (2, 0), (4, 1), (4, 4)
4,   7,       5:   (0, 0), (1, 1), (1, 6),
                   (6, 1), (6, 6)
5,  11,      11:   (0, 0), (1, 1), (1, 10),
                   (3, 0), (6, 2), (6, 9),
                   (7, 0), (9, 3), (9, 8),
                   (10, 1), (10, 10)
...

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.

Cf. A000040, A030205, A273163.

a(n) gives the number of solutions of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)), n >= 1.

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

A273163 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A272207(n).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, May 20 2016

The modularity pattern series is the expansion of the 39th modular cusp form of weight 2 and level N = 20, given in Table I of the Martin reference, i.e., eta^2(2*z)*eta^2(10*z) = Sum_{n >= 1} c(n)*q^n in powers of q = exp(2*Pi*i*z), with Im(z) > 0, where eta is the Dedekind function. c(prime(n)) = a(n), also for the bad primes 2 and 5 (the discriminant of this elliptic curve is -2^4*5^2). See also A030205 for the expansion in powers of q^2 (after deleting the factor q^(-1/2)): A030205((prime(n)-1)/2) = a(n), n >= 2.
See the comment on the Martin-Ono reference in A272207 which implies that eta^2(2*z) * eta^2(10*z) provides the modularity sequence for this elliptic curve.
The elliptic curve y^2 = x^3 + x^2 - x has the same p-defects.
For the above defined q-series coefficients c one seems to have for primes not 2 and 5 c(prime(n)^k) = (sqrt(prime(n)))^k*S(k, a(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. This corresponds to alpha-multiplicativity with alpha(x) = x (weight 2) except for primes 2 and 5, obtainable formally from Sum_{n>=1, without factors 2 or 5} c(n)/n^s = Product_{n=2,n>=4} 1/(1 - a(n)/prime(n)^s + prime(n)/prime(n)^{2*s}) (absolute convergence of the product seems to hold for Re(s) > 1 needed for the prime 3 factor). For alpha-multiplicativity see the Apostol reference, pp. 138-139 (exercise 6). c(2*k) = 0, c(2*k+1) = A030205(k), k >= 0. Thus c(2^k) = c(2)^k = 0, and it seems that c(5^k) = A030205((5^k-1)/2) = c(5)^k = (-1)^k, for k >= 1. Then one has multiplicativity of {c(n)} with the given c(p^k) formula.
For this multiplicity see the Michael Somos Oct 31 2005 comment on A030205, where c(n) is called b(n). - Wolfdieter Lang, May 24 2016
The Dirichlet sum could then be written as Sum_{n>=1} c(n)/n^s = 1/(1 - (-1)/5^s)* Product_{n=2,n>=4} 1/(1 - a(n)/prime(n)^s + prime(n)/prime(n)^{2*s}).

			{c(n)} multiplicativity test for n = 3^2*5^2 = 225: c(225) = A030205(112) = +1. c(3^2*5^2) = c(3^2)*c(5^2) = (3*S(2, a(2)/sqrt(3)))*(-1)^2 = ((-2)^2 - 3)*(+1) = +1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A030205, A049310, A272207.

a(n) = prime(n) - A272207(n), n >= 1, where A272207(n) is the number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)).

A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Number 62 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 - x^2 + x^6 + 2*x^7 - 2*x^10 + x^11 - x^12 - 2*x^15 + x^20 + ...
G.f. = q - q^5 - q^9 + q^25 + 2*q^29 - 2*q^41 + q^45 - q^49 - 2*q^61 + q^81 + ...

Bruce Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see page 44.

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

Cf. A030205, A030210, A159818.

Basis( CuspForms( Gamma1(80), 1), 413)[1]; /* Michael Somos, May 16 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^5], {x, 0, n}] (* Michael Somos, Aug 08 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/5, q^2] EllipticTheta[ 1, 2 Pi/5, q^2] / Sqrt[5], {q, 0, 4 n + 1}] // FullSimplify; (* Michael Somos, Aug 08 2011 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) * eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 04 2007 */

{a(n) = my(A, p, e, x, y); 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==5, (-1)^e, p%20>10, !(e%2), p%4==3, kronecker( -4, e+1), for( y=1, sqrtint(p\5), if( issquare(p - 5*y^2), x=y; break)); (-1)^(e*x) * (e+1))))}; /* Michael Somos, Sep 04 2007 */

Expansion of f(-x, -x^4) * f(-x^2, -x^3) in powers of x where f() is the Ramanujan two-variable theta function.
Expansion of q^(-1) * (phi(q) * phi(q^20) - phi(q^4) * phi(q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function.
Euler transform of period 5 sequence [ -1, -1, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 80^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (i^n +(-i)^n)/2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*y) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2. - Michael Somos, Sep 04 2007
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(5*k)).
a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n). - Michael Somos, May 16 2015
Convolution square is A030205. - Michael Somos, May 16 2015
a(n) = (-1)^n * A159818(n). - Michael Somos, May 16 2015

A159817 Coefficients of L-series for elliptic curve "80b2": y^2 = x^3 - x^2 - x.

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, -6, -6, -6, -20, -6, -4, -22, -12, 12, 10, 0, -18, -9, 4, -6, -2, -24

Offset: 0

Michael Somos, Apr 22 2009

sign

Number 61 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 - x^2 - 2*x^3 + x^4 + 2*x^6 - 2*x^7 - 6*x^8 + 4*x^9 - 4*x^10 + ...
G.f. = q + 2*q^3 - q^5 - 2*q^7 + q^9 + 2*q^13 - 2*q^15 - 6*q^17 + 4*q^19 + ...

Seiichi Manyama, 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. A030205, A159818.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] QPochhammer[ -x^5])^2, {x, 0, n}]; (* Michael Somos, Jun 10 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)^3 * eta(x^10 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)))^2, 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==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)))};

Expansion of (f(x) * f(x^5))^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)))^2 in powers of q.
Euler transform of period 20 sequence [ 2, -4, 2, -2, 4, -4, 2, -2, 2, -8, 2, -2, 2, -4, 4, -2, 2, -4, 2, -4, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, 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 / (80 t)) = 80 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)))^2.
a(n) = (-1)^n * A030205(n). Convolution square of A159818.

A276491 Expansion of qProduct_{k>=1} (1-q^(2k))^2(1-q^(10k))^2.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Sep 10 2016

Multiplicative. See A030205 for formula. - Andrew Howroyd, Aug 05 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A030205, A272207, A276030.

QPochhammer[x^2]^2*QPochhammer[x^10]^2 + O[x]^100 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2019 *)

seq(n)={Vec((eta(x^2 + O(x*x^n)) * eta(x^10 + O(x*x^n)))^2)} \\ Andrew Howroyd, Aug 05 2018

a(2n-1) = A030205(n-1), a(2n) = 0 for n > 0.
G.f.: (eta(x^2) * eta(x^10))^2. - Andrew Howroyd, Aug 05 2018
Euler transform of period 10 sequence [0, -2, 0, -2, 0, -2, 0, -2, 0, -4, ...]. - Georg Fischer, Nov 17 2022

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(5k))).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 51, 69, 96, 129, 175, 235, 312, 410, 539, 700, 913, 1173, 1508, 1923, 2450, 3105, 3920, 4926, 6177, 7710, 9614, 11923, 14766, 18218, 22435, 27550, 33750, 41231, 50278, 61150, 74259, 89932, 108744, 131193, 158025, 189979, 227998, 273125, 326692

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000712 and A145466.
Convolution inverse of A030202.
Number of partitions of n if there are 2 kinds of parts that are multiples of 5.

			a(5) = 8 because we have [5], [5'], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000712, A002513, A030202, A030205, A145466, A318026, A318027.

a:=series(mul(1/((1-x^k)*(1-x^(5*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(5 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^5]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + x^(3 k) + 2 x^(4 k))/(k (1 - x^(5 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 5 k], {k, 0, n/5}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k) + 2*x^(4 k))/(k*(1 - x^(5*k)))).

a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A273163 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A272207(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A159817 Coefficients of L-series for elliptic curve "80b2": y^2 = x^3 - x^2 - x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A276491 Expansion of q*Product_{k>=1} (1-q^(2*k))^2*(1-q^(10*k))^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(5*k))).

Views

Author

Keywords

Comments

Examples

Links

A276491 Expansion of qProduct_{k>=1} (1-q^(2k))^2(1-q^(10k))^2.

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(5k))).