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Expansion of eta(q) * eta(q^4) * eta(q^14)^4 / (eta(q^2) * eta(q^7) * eta(q^28)) in powers of q.
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%I #28 Sep 08 2022 08:45:44

%S 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,

%T 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,

%U 0,-1,-1,-2,0,0,0,0,4,2,-3,0,2,0,0,2,2,-2,0

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

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

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

%H G. C. Greubel, <a href="/A159813/b159813.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F 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

%F 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

%F a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. a(7*n) = A215556(n). - _Michael Somos_, Aug 15 2012

%e 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 + ...

%t 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 *)

%t 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 *)

%t 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 *)

%t 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 *)

%o (Magma) Basis(CuspidalSubspace(HalfIntegralWeightForms(28,3/2)),100)

%o (Magma) A := Basis( CuspForms( Gamma1(28), 3/2), 81); A[1] - A[2]; /* _Michael Somos_, Sep 06 2015 */

%o (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 */

%Y Cf. A215556.

%K sign

%O 1,15

%A _Steven Finch_, Apr 22 2009