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A212318
Expansion of phi(q^2)^2 / (phi(-q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.
7
1, 2, 8, 16, 32, 60, 96, 160, 256, 394, 624, 944, 1408, 2092, 3008, 4320, 6144, 8612, 12072, 16720, 22976, 31424, 42528, 57312, 76800, 102254, 135728, 179104, 235264, 307852, 400704, 519808, 671744, 864672, 1109904, 1419456, 1809568, 2300284, 2914272, 3682400
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 chi(q^2)^5 / (chi(-q) * chi(q^4))^2 in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q^4)^12 * eta(q^16)^2 / (eta(q)^2 * eta(q^2)^3 * eta(q^8)^9) in powers of q.
Euler transform of period 16 sequence [ 2, 5, 2, -7, 2, 5, 2, 2, 2, 5, 2, -7, 2, 5, 2, 0, ...].
a(n) = 2 * A215348(n) unless n=0. a(2*n) = A014969(n). a(2*n + 1) = 2 * A232772(n).
a(n) ~ exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
EXAMPLE
G.f. = 1 + 2*q + 8*q^2 + 16*q^3 + 32*q^4 + 60*q^5 + 96*q^6 + 160*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}]
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^12 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^8 + A)^9), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 25 2013
STATUS
approved