A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.
1, -9, 27, -9, -117, 216, 27, -450, 459, -9, -648, 1080, -117, -1530, 1350, 216, -1845, 2592, 27, -3258, 2808, -450, -3240, 4752, 459, -5409, 4590, -9, -5850, 7560, -648, -8658, 7371, 1080, -7776, 10800, -117, -12330, 9774, -1530, -11016, 15120, 1350, -16650
Offset: 0
Keywords
Examples
G.f. = 1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + ...
References
- G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 313, Equ. (14.2.13).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- George E. Andrews and Bruce C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
- Jonathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. See p. 697.
- David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
- Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
- Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Programs
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Magma
A := Basis( ModularForms( Gamma1(3), 3), 44); A[1] - 9*A[2]; /* Michael Somos, May 18 2015 */
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Mathematica
a[ n_] := If[ n < 1, Boole[ n == 0], - 9 DivisorSum[ n, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Jul 19 2012 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
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PARI
{a(n) = if( n<1, n==0, -9 * sumdiv( n, d, d^2 * kronecker(-3, d)))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^9 / eta(x^3 + A)^3, n))};
Formula
G.f.: Product_{k>0} (1 - x^k)^9 / (1 - x^3)^3 = 1 - 9 * Sum_{k>0} x^k * (1 - x^k -6 * x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3.
Expansion of b(q)^3 in powers of q where b() is a cubic AGM theta function.
Euler transform of period 3 sequence [ -9, -9, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u*w * (u + 6*v - 8*w).
Given A = A0 + A1 + A2 is the 3-section, then 0 = A1^3 + A2^3 - 3*A0*A1*A2. A0 = A(q^3) = b(q^3)^3, A1 = -3 * a(q^3)^2 * c(q^3), A2 = 3 * a(q^3) * c(q^3)^2 where a(), b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 19683^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106402. - Michael Somos, Mar 11 2012
A008654(n) = a(n) + 27 * A106402(n) is the identity a(q)^3 = b(q)^3 + c(q)^3. - Michael Somos, Jul 19 2012
a(n) = -9 * b(n) where b(n) is multiplicative with a(0) = 1, b(p^e) = 1, if p=3, b(p^e) = b(p) * b(p^(e-1)) + Kronecker(-3, p) * p^2 * b(p^(e-2)) otherwise. - Michael Somos, May 18 2015
Convolution cube of A005928. - Michael Somos, May 18 2015
Comments