OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Since phi(-x) = 1 + 2*Sum_{k >= 1} (-1)^k*x^(k^2) == 1 (mod 2), it follows that the g.f. psi(x^4) / phi(-x) == psi(x^4) == Sum_{k >= 0} x^(2*k*(k+1)) (mod 2). Hence a(n) is odd iff n = 2*k*(k + 1) for some nonegative integer k. - Peter Bala, Jan 07 2025
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * eta(q^2) * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, 2, 2, 1, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A093085.
a(n) ~ exp(sqrt(n)*Pi)/(16*n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
EXAMPLE
1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 44*x^6 + 72*x^7 + 114*x^8 + ...
q + 2*q^3 + 4*q^5 + 8*q^7 + 15*q^9 + 26*q^11 + 44*q^13 + 72*q^15 + 114*q^17 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, q^2]/(2*Sqrt[q]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[A187154[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Michael Somos, Mar 08 2011
STATUS
approved