A252650 Expansion of (eta(q) * eta(q^2) * eta(q^3) / eta(q^6))^2 in powers of q.
1, -2, -3, 4, 6, 6, -12, -16, -3, 4, 36, 12, -12, -28, -24, 24, 6, 18, -12, -40, -18, 32, 72, 24, -12, -62, -42, 4, 48, 30, -72, -64, -3, 48, 108, 48, -12, -76, -60, 56, 36, 42, -96, -88, -36, 24, 144, 48, -12, -114, -93, 72, 84, 54, -12, -144, -24, 80, 180
Offset: 0
Keywords
Examples
G.f. = 1 - 2*q - 3*q^2 + 4*q^3 + 6*q^4 + 6*q^5 - 12*q^6 - 16*q^7 - 3*q^8 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- C. Kassel and C. Reutenauer, On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus, arXiv:1505.07229 [math.AG], 2015, see page 31 7.2(d).
- Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Magma
A := Basis( ModularForms( Gamma0(36), 2), 58); A[1] - 2*A[2] - 3*A[3] + 4*A[4] + 6*A[5] + 6*A[6] - 12*A[7] - 16*A[8] - 3*A[9] + 4*A[10] + 36*A[11] - 12*A[12];
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Mathematica
QP = QPochhammer; s = (QP[q]*QP[q^2]*(QP[q^3]/QP[q^6]))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
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PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) / eta(x^6 + A))^2, n))};
Formula
Expansion of f(-q)^4 * f(q, q^2)^2 / f(-q^3)^2 = f(-q)^4 * f(-q^6)^2 / f(-q, -q^5)^2 in powers of q where f() is a Ramanujan theta function.
Expansion of b(q) * c(q) * sqrt(b(q^2) / (3 * c(q^2))) in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [-2, -4, -4, -4, -2, -4, ...].
G.f.: Product_{k>0} (1 - x^k)^4 / (1 - x^k + x^(2*k))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098098.
Comments