A256574 Expansion of chi(x) * psi(-x^3) * psi(x^48) in powers of x where psi(), chi() are Ramanujan theta functions.
1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0
Offset: 0
Keywords
Examples
G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^48 + x^49 + ... G.f. = q^19 + q^22 + q^34 + q^43 + q^67 + q^82 + q^118 + q^139 + q^163 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..86 from Michael Somos)
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, 0, x^24] / (2^(3/2) x^(51/8)), {x, 0, n}]; a[ n_] := If[ n < 0 || Mod[n, 8] == 2, 0, (1/2) Times @@ (Which[# < 5, Boole[# + #2 == 3], Mod[#, 8] > 4, Mod[#2 + 1, 2], True, #2 + 1] & @@@ FactorInteger[ 3 n + 19])]; (* Michael Somos, Oct 25 2015 *) -
PARI
{a(n) = my(A, p, e); if( n<0 || n%8 == 2, 0, A = factor(3*n + 19); 1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^96 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^48 + A)), n))};
Formula
Expansion of q^(-19/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) * eta(q^96)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^48)) in powers of q.
Euler transform of a period 96 sequence.
2 * a(n) = A257403(3*n + 19) unless n == 2 (mod 8).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = a(16*n + 9) = a(16*n + 13) = 0.
a(n) = (-1)^n * A255320(n). - Michael Somos, Apr 24 2015
Expansion of f(x, x^5) * psi(x^48) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Oct 25 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263767.
Comments