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A132965
Expansion of f(-q^8) * chi(q)^2 in powers of q where f(), chi() are Ramanujan theta functions.
4
1, 2, 1, 2, 4, 4, 5, 6, 8, 10, 12, 14, 17, 22, 24, 30, 36, 40, 48, 56, 65, 76, 88, 100, 116, 134, 152, 174, 200, 226, 257, 292, 328, 372, 420, 472, 532, 598, 668, 750, 840, 936, 1045, 1166, 1296, 1442, 1604, 1776, 1972, 2186, 2416, 2672, 2952, 3256, 3592, 3960
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^8) / (eta(q)^2 * eta(q^4)^2) in powers of q.
Euler transform of period 8 sequence [ 2, -2, 2, 0, 2, -2, 2, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132966.
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)).
a(n) ~ exp(sqrt(n)*Pi/2) / (4*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
G.f. = q + 2*q^5 + q^9 + 2*q^13 + 4*q^17 + 4*q^21 + 5*q^25 + 6*q^29 + 8*q^33 + ...
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[(1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)^2), n))};
CROSSREFS
Cf. A132966.
Sequence in context: A153898 A108802 A023673 * A365005 A022597 A073252
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 23 2007
STATUS
approved