A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
1, -1, 2, -2, 1, -2, 0, -2, 0, 0, 2, 0, 3, -1, 2, -2, 2, -4, 0, 0, 0, 0, 2, 0, 3, 0, 2, -4, 0, -2, 0, -2, 0, 0, 0, 0, 2, -3, 4, -2, 1, -2, 0, -2, 0, 0, 2, 0, 2, -2, 2, -2, 4, -2, 0, 0, 0, 0, 0, 0, 3, 0, 4, -2, 0, -2, 0, -2, 0, 0, 0, 0, 4, -3, 2, -2, 0, -4, 0
Offset: 0
Keywords
Examples
G.f. = 1 - x + 2*x^2 - 2*x^3 + x^4 - 2*x^5 - 2*x^7 + 2*x^10 + 3*x^12 - x^13 + ... G.f. = q - q^3 + 2*q^5 - 2*q^7 + q^9 - 2*q^11 - 2*q^15 + 2*q^21 + 3*q^25 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
- 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
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Programs
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Mathematica
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Jun 09 2015 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ -x^3] / (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^12]), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
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PARI
{a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, kronecker( -6, d)))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^4 * eta(x^24 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^3), n))}; -
PARI
{a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, (-1)^e, p%24 < 12, (e+1) * if( p%24 < 6, 1, (-1)^e), (1 + (-1)^e) / 2 )))};
Formula
Expansion of phi(-x^3) * psi(x^4) - x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q) * eta(q^4)^4 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3) * eta(q^8) * eta(q^12)^3) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, 2, 0, -2, -1, -1, -1, -1, 0, 2, -1, -2, -1, 2, 0, -1, -1, -1, -1, -2, 0, 2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker( 6, k) * q^k / (1 + q^(2*k)) = Sum_{k>=0} a(k) * q^(2*k + 1).
G.f.: Product_{k>0} (1 + (-x)^k) * (1 - (-x^2)^k) * (1 - (-x^3)^k) * (1 + (-x^6)^k).
a(n) = (-1)^n * A129402(n). a(3*n + 1) = -a(n). a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
a(12*n) = A113700(n). a(12*n + 2) = 2 * A128583(n). a(12*n + 5) = -2 * A128591(n). - Michael Somos, Jun 09 2015
a(n) = (-1)^floor(n/2) * A128580(n) = (-1)^(n + floor(n/2)) * A134177(n). - Michael Somos, Jul 29 2015
a(4*n) = A260308(n). a(4*n + 1) = - A257920(n). a(4*n + 2) = 2 * A259895(n). a(4*n + 3) = -2 * A259896(n). - Michael Somos, Jul 29 2015
a(12*n + 3) = -2 * A260089(n). - Michael Somos, Jul 29 2015
Comments