A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
1, -1, 0, -1, 2, 0, 0, -1, 2, -2, 0, 0, 2, 0, 0, -1, 2, -2, 0, -2, 0, 0, 0, 0, 3, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -2, 2, 0, 0, -2, 2, 0, 0, 0, 4, 0, 0, 0, 1, -3, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -2, 2, -2, 0, 0, 0
Offset: 1
Examples
G.f. = q - q^2 - q^4 + 2*q^5 - q^8 + 2*q^9 - 2*q^10 + 2*q^13 - q^16 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..2500
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ 1/2 - (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^9]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 02 2015 *) -
PARI
{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==3, if( p>3, 1, 2) * (1-e%2), e+1)))};
Formula
Expansion of 1/2 - (eta(q)^4 * eta(q^18)^2 + eta(q^2)^2 * eta(q^9)^4) / (2 * eta(q^2) * eta(q^18))^2 in powers of q. - Michael Somos, Jun 02 2015
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 1 + (-1)^e if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(9*n) = 2 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.
-2 * a(n) = A258322(n) unless n = 0 or n == 2 (mod 3).
Sum_{k=1..n} abs(a(k)) ~ (5*Pi/18) * n. - Amiram Eldar, Jan 29 2024
Comments