A159813 Expansion of eta(q) * eta(q^4) * eta(q^14)^4 / (eta(q^2) * eta(q^7) * eta(q^28)) in powers of q.

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

Offset

1,15

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Unique cusp form of weight 3/2, level 28 and trivial character.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * psi(-q) * psi(-q^7) * phi(q^7) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2012

Euler transform of period 28 sequence [ -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, -3, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -3, ...]. - Michael Somos, Aug 15 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 56^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215556. - Michael Somos, Aug 15 2012

a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. a(7*n) = A215556(n). - Michael Somos, Aug 15 2012

Example

G.f. = q - q^2 - q^4 + q^7 + q^8 - q^9 + q^14 - 2*q^15 + q^16 + 3*q^18 - 2*q^21 + ...

Mathematica

max = 100; s28 = Table[{-1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, -3, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -3}, {max/28 // Ceiling}] // Flatten; coes = Series[ 1 + Sum[a[n]*x^n, {n, 1, max}] - Product[1/(1 - x^n)^s28[[n]], {n, 1, max}], {x, 0, max}] // CoefficientList[#, x] &; sol = Solve[Thread[coes == 0]]; Join[{1}, Table[a[n], {n, 1, max}] /. sol // First] (* Jean-François Alcover, Jun 20 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(7/2)] EllipticTheta[ 3, 0, q^7]/2 , {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) q^(7/8) EllipticTheta[ 2, Pi/4, q^(1/2)] QPochhammer[ -q^7] QPochhammer[ q^14], {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)
a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^{1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 3}[[Mod[k, 28, 1]]], {k, n - 1}], {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)

Prog

(Magma) Basis(CuspidalSubspace(HalfIntegralWeightForms(28, 3/2)), 100)
(Magma) A := Basis( CuspForms( Gamma1(28), 3/2), 81); A[1] - A[2]; /* Michael Somos, Sep 06 2015 */
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^14 + A)^4 / (eta(x^2 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Aug 15 2012 */

Crossrefs

Cf. A215556.

Sequence in context: A136255 A194812 A305320 * A157409 A245960 A340867

Adjacent sequences: A159810 A159811 A159812 * A159814 A159815 A159816

Keyword

sign

Author

Steven Finch, Apr 22 2009

Status

approved