A115155 Expansion of (eta(q^3) * eta(q^5))^3 + (eta(q) * eta(q^15))^3 in powers of q.
1, 1, -3, -3, 5, -3, 0, -7, 9, 5, 0, 9, 0, 0, -15, 5, -14, 9, -22, -15, 0, 0, 34, 21, 25, 0, -27, 0, 0, -15, 2, 33, 0, -14, 0, -27, 0, -22, 0, -35, 0, 0, 0, 0, 45, 34, -14, -15, 49, 25, 42, 0, -86, -27, 0, 0, 66, 0, 0, 45, -118, 2, 0, 13, 0, 0, 0, 42, -102, 0, 0
Offset: 1
Examples
G.f. = q + q^2 - 3*q^3 - 3*q^4 + 5*q^5 - 3*q^6 - 7*q^8 + 9*q^9 + 5*q^10 +...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005, see page 5. [Cached copy, with permission of the author]
- M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures. see p. 26 f(q)
Programs
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Magma
A := Basis( CuspForms( Gamma1(15), 3), 80); A[1] + A[2] - 3*A[3] - 3*A[4] + 5*A[5] - 3*A[6] - 7*A[8]; /* Michael Somos, Oct 13 2015 */
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Mathematica
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] QPochhammer[ q^5])^3 + q^2 (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* Michael Somos, May 24 2013 *) -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^3 + x * (eta(x + A) * eta(x^15 + A))^3, n))}; -
Sage
A = CuspForms( Gamma1(15), 3, prec=80) . basis(); A[0] + A[1] - 3*A[2] - 3*A[3] + 5*A[4] - 3*A[5] - 7*A[7]; # Michael Somos, May 28 2013
Formula
a(n) is multiplicative with a(3^e) = (-3)^e, a(5^e) = 5^e, a(p^e) = p^e if e even else 0 if p == 7, 11, 13, 14 (mod 15), a(p^e) = a(p) * a(p^(e-1)) - p^2 * a(p^(e-2)) if p == 1, 2, 4, 8 (mod 15).
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 23 2012
a(2^n) = A106853(n).
Comments