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A258096
Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.
4
1, 2, 4, 8, 7, 10, 12, 8, 18, 18, 16, 24, 21, 20, 28, 32, 20, 32, 36, 24, 42, 42, 28, 48, 57, 36, 52, 40, 36, 58, 60, 56, 48, 66, 48, 72, 74, 42, 80, 80, 61, 82, 72, 56, 90, 96, 64, 72, 98, 70, 100, 104, 64, 106, 108, 72, 114, 96, 84, 144, 111, 84, 104, 128
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^(-1/2) * eta(q^2) * eta(q^4)^7 / (eta(q)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -6, 2, 1, 2, -4, ...].
a(n) = (-1)^n * A209940(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 7*x^4 + 10*x^5 + 12*x^6 + 8*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 7*q^9 + 10*q^11 + 12*q^13 + 8*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4]^7 / (QPochhammer[ x]^2 QPochhammer[ x^8]^2), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] EllipticTheta[ 4, 0, x^4]^4 / (EllipticTheta[ 4, 0, x] 2 x^(1/2)), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^7 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 19 2015
STATUS
approved