A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.
1, -2, 0, 0, 2, 0, 1, -2, 0, -2, 2, 0, 0, 0, 0, -2, 2, 0, 1, -2, 0, 0, 4, 0, 0, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 3, -2, 0, 0, 2, 0, 2, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 2, -2, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, -2, 0, -2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0
Offset: 0
Keywords
Examples
G.f. = 1 - 2*x + 2*x^4 + x^6 - 2*x^7 - 2*x^9 + 2*x^10 - 2*x^15 + 2*x^16 + ... G.f. = q^3 - 2*q^7 + 2*q^19 + q^27 - 2*q^31 - 2*q^39 + 2*q^43 - 2*q^63 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
-
Mathematica
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 3, KroneckerSymbol[ -3, #] &]]; a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0, n}]; a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2 + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 3))]; (* Michael Somos, Oct 01 2015 *) -
PARI
{a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 3, d, kronecker(-3, d)))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))}; -
PARI
{a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 3); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))}; /* Michael Somos, Oct 01 2015 */
Formula
Expansion of q^(-3/4) * eta(q)^2 * eta(q^12)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = A262780(2*n + 1). - Michael Somos, Oct 01 2015
Comments