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A246835
Expansion of psi(-x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
3
1, -2, 3, -6, 4, -4, 7, -2, 8, -10, 4, -10, 9, -6, 8, -10, 4, -8, 16, -8, 9, -12, 8, -12, 20, -6, 8, -10, 8, -18, 11, -12, 8, -20, 12, -8, 20, -6, 20, -26, 8, -8, 15, -10, 16, -18, 12, -16, 20, -10, 16, -16, 8, -24, 24, -8, 21, -26, 8, -20, 20, -14, 8, -28
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 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.
a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(n).
EXAMPLE
G.f. = 1 - 2*x + 3*x^2 - 6*x^3 + 4*x^4 - 4*x^5 + 7*x^6 - 2*x^7 + 8*x^8 + ...
G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};
CROSSREFS
KEYWORD
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AUTHOR
Michael Somos, Sep 04 2014
STATUS
approved