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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.

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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

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

Cf. A030210, A049310, A159817, A273163.

Programs

  • Magma
    Basis( CuspForms( Gamma0(20), 2), 145) [1]; /* Michael Somos, May 27 2014 */
  • Magma
    A := Basis( CuspForms( Gamma1(20), 2), 145); A[1] - 2*A[3]; /* Michael Somos, May 17 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, May 28 2013 *)
  • PARI
    {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 */
  • PARI
    {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 */
  • PARI
    {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 */
  • PARI
    {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 */
  • Sage
    CuspForms( Gamma0(20), 2, prec=92).0; # Michael Somos, May 28 2013

Formula

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

A159818 Expansion of f(q) * f(q^5) in powers of q where f() is a Ramanujan theta function.

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

Views

Author

Michael Somos, Apr 22 2009

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 71 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			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 + ...

Links

Crossrefs

Cf. A030202, A159817.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
  • PARI
    {a(n) = my(A, p, e, x, z); 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, 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), z = if(x%2, y, x)%4/2; break)); (-1)^(e*z) *(e+1))))};
  • PARI
    {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)), n))};

Formula

Expansion of q^(-1/4) * eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, -4, 1, -1, 1, -2, 2, -1, 1, -2, 1, -2, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*z) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2 and z = 1 if x or y == 0 (mod 4) else z = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = (320)^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)).
a(n) = (-1)^n * A030202(n). Convolution square is A159817. a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n).
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