OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001318.
The exponents in the q-series for this sequence are the squares of the numbers of A007310.
Number 14 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016
LINKS
Seiichi Manyama, 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
FORMULA
Expansion of phi(x^3) / chi(x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q) * eta(q^4) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [ -1, 1, 1, 0, -1, -2, -1, 0, 1, 1, -1, -1, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2*e)) = +1 if p == 1, 7 (mod 8) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 4 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133988.
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -a(n). a(n) = (-1)^n * A080995(n).
G.f. Sum_{k>=0} a(k) * q^(24*k + 1) = Sum_{k in Z} (-1)^floor(k/2) * q^(6*k + 1)^2.
Expansion of f(-x^5, -x^7) - x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015
EXAMPLE
G.f. = 1 - x + x^2 - x^5 - x^7 + x^12 - x^15 + x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 - q^169 + q^289 - q^361 + q^529 + q^625 + ...
MATHEMATICA
a[ n_] := (-1)^n Boole[ IntegerQ[ Sqrt[24 n + 1]]]; (* Michael Somos, Jan 10 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
PROG
(PARI) {a(n) = (-1)^n * issquare( 24*n + 1) };
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 01 2007, Oct 04 2007
STATUS
approved