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A261426
Expansion of f(-x^3)^3 * phi(x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
5
1, 1, 2, 0, 2, 1, 4, 2, 5, 2, 6, 2, 6, 0, 4, 4, 7, 2, 4, 0, 6, 1, 8, 4, 4, 4, 10, 2, 8, 2, 12, 4, 8, 5, 6, 0, 14, 2, 8, 2, 11, 6, 6, 4, 8, 2, 8, 4, 8, 6, 14, 0, 6, 0, 12, 6, 15, 4, 14, 2, 14, 4, 8, 8, 12, 7, 14, 0, 12, 2, 16, 10, 8, 4, 10, 6, 14, 0, 16, 4, 16
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1/3) * q^(-1/3) * c(q) * phi(q^6) in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM function. - Michael Somos, Sep 01 2015
Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^12)^5 / (eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -5, 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (128/3)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261426.
24 * a(n) = A261394(6*n + 2).
a(n) = A261444(2*n). Michael Somos, Sep 01 2015
EXAMPLE
G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 5*x^8 + 2*x^9 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 4*q^19 + 2*q^22 + 5*q^25 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 3, 0, x^6] / QPochhammer[ x], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 18 2015
STATUS
approved