cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Number 49 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).
Glaisher (1884) essentially defines chi(n) as the sum over all solutions of n = x^2 + y^2 with even y and nonnegative odd x of x * (-1)^((x + y - 1)/2) and proves that it is multiplicative. If n is not == 1 (mod 4) then chi(n) = 0. - Michael Somos, Jun 18 2012
Denoted by g_2(q) in Cynk and Hulek on page 8 as the unique weight 2 level 32 newform. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012
The weight 2 level N = 32 newform (eta(q^4)*eta(q^8))^2 belongs to the elliptic curves y^2 = x^3 + 4*x , y^2 = x^3 - x, y^2 = x^3 - 11*x - 14 and y^2 = x^3 - 11*x + 14. See the Martin-Ono link, Theorem 2, row N = 32, and the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 26 2016

Examples

			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 + ...
G.f. (eta(q^4)*eta(q^8))^2 = q - 2*q^5 - 3*q^9 + 6*q^13 + 2*q^17 - q^25 - 10*q^29 - 2*q^37 + 10*q^41 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000203, A002172, A278720, A279955.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* Michael Somos, Jun 12 2014 */
  • Magma
    qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* Michael Somos, Jun 12 2014 */
  • Magma
    Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* Michael Somos, Mar 25 2015 */
  • Mathematica
    max=100; f[x_] := Product[(1-x^k)*(1-x^(2k)), {k, 1, max}]^2; CoefficientList[ Series[ f[x], {x, 0, max}], x](* Jean-François Alcover, Jan 04 2012, after g.f. *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
  • PARI
    {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* Michael Somos, Jul 27 2006 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^2, n))}; /* Michael Somos, Jul 27 2006 */
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor( 4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, (-p)^(e/2) * (1 + (-1)^e) / 2, forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0 = 1; y = a1 = x * (-1)^((x + y)\2) * if(y, 2, 1); for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1 )))}; /* Michael Somos, Jun 18 2012 */

Formula

Expansion of (psi(x) * phi(-x))^2 = phi(-x) * f(-x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q)^2 * eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [-2, -4, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) and b(p) = p - number of solutions of y^2 = x^3 - x (mod p). - Michael Somos, Jul 27 2006. b(p(n)) = A278720(n). - Wolfdieter Lang, Dec 26 2016
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^2.
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (Sum_{k>=0} (-1)^k * (2*k + 1) * x^(2*k + 1)^2) * (Sum_{k in Z} (-1)^k * x^(4*k)^2).
Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

Original entry on oeis.org

1, -6, 9, 10, -30, 0, 11, 42, 0, -70, 18, -54, 49, 90, 0, -22, -60, 0, -110, 0, 81, 180, -78, 0, 130, -198, 0, -182, -30, 90, 121, 84, 0, 0, 210, 0, -252, -102, -270, 170, 0, 0, -69, 330, 0, -38, 420, 0, -190, -390, 0, -108, 0, 0, 0, -300, 99, 442, 210, 0, 418, -294, 0, 0, -510, 378, -540, 138, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This is Glaisher's function lambda(m). It appears to be defined only for odd m, and lambda(4t-1) = 0 (t >= 1), lambda(4t+1) = a(t) (t >= 0). - N. J. A. Sloane, Nov 25 2018
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 36 of the 74 eta-quotients listed in Table I of Martin (1996).
Dickson, v.2, p. 295 briefly states a result of Glaisher, 1883, pp 212-215. This result is that a(n) is the sum over all solutions of 16*n + 4 = x^2 + y^2 + z^2 + w^2 in nonnegative odd integers of chi(x) and is also the sum over all solutions of 8*n + 2 = x^2 + y^2 in nonnegative odd integers of chi(x) * chi(y) where chi(x) = x if x == 1 (mod 4) and -x if x == 3 (mod 4). [Michael Somos, Jun 18 2012]
Denoted by g_3(q) in Cynk and Hulek on page 8 as the unique weight 3 Hecke eigenform of level 16 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012

Examples

			G.f. = 1 - 6*x + 9*x^2 + 10*x^3 - 30*x^4 + 11*x^6 + 42*x^7 - 70*x^9 + 18*x^10 + ...
G.f. = q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...

References

  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 295, and vol. 3, p. 134.
  • 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 page 340.
  • J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m), Quart. J. Math, 37 (1906), 36-48.
  • Morris Newman, A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(16), 3), 274); A[2] - 6*A[6] + 9*A[10] + 10*A[14] - 30*A[18]; /* Michael Somos, May 17 2015 */
  • Magma
    A := Basis( CuspForms( Gamma1(16), 3), 274); A[1] - 6*A[5]; /* Michael Somos, Jan 09 2017 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/16 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q]^4 EllipticTheta[ 3, 0, q], {q, 0, 4 n + 1}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[ 16 n + 4]}, SeriesCoefficient[ Sum[ Mod[k, 2] q^k^2, {k, m}]^3 Sum[ KroneckerSymbol[ -4, k] k q^k^2, {k, m}], {q, 0, 16 n + 4}]]]; (* Michael Somos, Jun 12 2012 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^3 / (4 q^(1/2)), {q, 0, 2 n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/4) EllipticThetaPrime[ 1, -Pi/4, q] EllipticTheta[ 1, -Pi/4, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/16) EllipticThetaPrime[ 1, 0, q] EllipticTheta[ 1, -Pi/2, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6, n))};
  • PARI
    {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==3, if( e%2, 0, p^e), forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 21 2006 */
  • PARI
    {a(n)=local(tn=(sqrtint(8*n+1)+1)\2);polcoeff(sum(m=0,tn,(1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0,tn,sum(k=1,tn,(1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))),n)} /* Paul D. Hanna, Mar 15 2010 */

Formula

Expansion of q^(-1/4)/16 * theta_2(q)^4 * theta_3(q) * theta_4(q) in powers of q. - [Dickson, v. 3, p. 134] from Stieltjes footnote 160. Michael Somos, Jun 18 2012
Expansion of q^(-1/2) / 4 * k * k' * (K / (pi/2))^3 in powers of q^2 where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
G.f.: Product_{k>0}(1 - x^k)^6.
Given g.f. A(x), then A(q^4) = f(-q^4)^6 = phi(q) * phi(-q) * psi(q^2)^4 where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 23 2006
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^2 if p == 1 (mod 4) and b(p) = 2 * (x^2 - y^2) where p = x^2 + y^2 and y is even. - Michael Somos, Aug 23 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (1/2) * Sum_{u,v in Z} (u*u - 4*v*v) * x^(u*u + 4*v*v). - Michael Somos, Jun 14 2007
G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [Paul D. Hanna, Mar 15 2010]
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M be a positive integer whose prime factors are all congruent to 3 (mod 4) - see A004614. Then a( M^2*n + (M^2 - 1)/4 ) = M^2*a(n). See Cooper et al., equation 5. - Peter Bala, Dec 01 2020
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = ((x+y*i)^(2*e+2) - (x-y*i)^(2*e+2))/((x+y*i)^2 - (x-y*i)^2) if p == 1 (mod 4) where p = x^2 + y^2 and x is odd. - Jianing Song, Mar 19 2022

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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).

Examples

			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)

References

  • 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)

Links

Crossrefs

Cf. A002607, A215472, A247067.

Programs

  • Magma
    Basis( CuspForms( Gamma1(4), 5), 71) [1]; /* Michael Somos, May 27 2014 */
  • Mathematica
    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 *)
  • PARI
    {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 */
  • PARI
    {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 */
  • PARI
    {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 */
  • Sage
    CuspForms( Gamma1(4), 5, prec=71).0; # Michael Somos, May 28 2013

Formula

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

A215601 Expansion of phi(-x)^2 * f(-x)^6 + 32 * x * psi(-x)^2 * f(-x^4)^6 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 22, -27, -18, -94, 0, 359, -130, 0, 214, -230, -594, -343, 518, 0, 830, -396, 0, 1098, 0, 729, -2068, -1670, 0, 594, 598, 0, -1746, 2002, 486, -1331, 5148, 0, 0, -1606, 0, -2860, -3514, 2538, 286, 0, 0, -1873, -4082, 0, 3942, 4708, 0, 5362, 1174, 0, -5060
Offset: 0

Views

Author

Michael Somos, Aug 16 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by g_4(q) in Cynk and Hulek on page 8 as the unique weight 4 Hecke eigenform of level 32 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

Examples

			G.f. = 1 + 22*x - 27*x^2 - 18*x^3 - 94*x^4 + 359*x^6 - 130*x^7 + 214*x^9 - 230*x^10 + ..
G.f. = q + 22*q^5 - 27*q^9 - 18*q^13 - 94*q^17 + 359*q^25 - 130*q^29 + 214*q^37 + ...

Links

Crossrefs

Cf. A000729, A002171, A008441, A215472, A215600.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2 + 32 x (QPochhammer[ x] QPochhammer[ x^4]^4 / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jan 11 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^5 / eta(x^2 + A) )^2 + 32 * x * ( eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A) )^2, n))};
  • PARI
    {a(n) = local(A, p, e, x, y, a0, a1, w=3); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4==3, if( e%2, 0, (-p)^(w*e/2)), y=-sum( i=0, p-1, kronecker( i^3-i, p)); a0=2; a1=y; for( i=2, w, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for( i=2, e, x=y*a1 -p^w*a0; a0=a1; a1=x); a1)))))};

Formula

Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 + 32 * (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^10 (t/i)^4 f(t) where q = exp(2 Pi i t).
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -27 * a(n). a(n) = A215600(2*n).

A258739 Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -82, -243, -1194, 2242, 0, 3599, 2950, 0, -12242, -20950, 19926, -16807, 7294, 0, 18950, 97908, 0, -88806, 0, 59049, -183844, 51050, 0, -92142, -98002, 0, 246486, 118706, 290142, -161051, -38868, 0, 0, 75658, 0, -241900, 47614, -544806, -493658, 0, 0
Offset: 0

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Author

Michael Somos, Jun 08 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
Denoted by g_6(q) in Cynk and Hulek on page 8 as a level 32 cusp form of weight 6.

Examples

			G.f. = 1 - 82*x - 243*x^2 - 1194*x^3 + 2242*x^4 + 3599*x^6 + 2950*x^7 + ...
G.f. = q - 82*q^5 - 243*q^9 - 1194*q^13 + 2242*q^17 + 3599*q^25 + 2950*q^29 + ...

Links

Crossrefs

Cf. A000729, A002171, A008441, A215472, A215601.

Programs

  • Magma
    A := Basis( CuspForms( Gamma0(32), 6), 165); A[1]  - 82*A[5] - 243*A[9] - 1194*A[13] + 2242*A[16];
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^3 / QPochhammer[ x^2])^6 - 64 x (QPochhammer[ x^2]^3 / QPochhammer[ x])^6, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^2 + A))^6 - 64 * x * (eta(x^2 + A)^3 / eta(x + A))^6, n))};
  • PARI
    {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==3, if(e%2, 0, (-p)^(5*e/2)), y = -sum(i=0, p-1, kronecker(i^3-i, p)); a0=2; a1=y; for(i=2, 5, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for(i=2, e, x=y*a1 -p^5*a0; a0=a1; a1=x); a1)))};

Formula

Expansion of q^(-1/4) * ((eta(-q)^3 / eta(-q^2))^6 - 64 * (eta(-q^2) / eta(-q))^6) in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(5*e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = -(32^3) (t/i)^6 f(t) where q = exp(2 Pi i t).
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