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A215348
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Expansion of q * phi(q) * psi(q^8) / (phi(-q) * phi(q^4)) in powers of q where phi(), psi() are Ramanujan theta functions.
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7
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1, 4, 8, 16, 30, 48, 80, 128, 197, 312, 472, 704, 1046, 1504, 2160, 3072, 4306, 6036, 8360, 11488, 15712, 21264, 28656, 38400, 51127, 67864, 89552, 117632, 153926, 200352, 259904, 335872, 432336, 554952, 709728, 904784, 1150142, 1457136, 1841200, 2320128
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OFFSET
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1,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of q * (f(q) * f(-q^16) / (f(-q) * f(q^4)))^2 = q * (chi(-q^2) * chi(-q^4) / (chi(-q) * chi(-q^8))^2)^2 in powers of q where chi(), f() are Ramanujan theta functions.
Expansion of (eta(q^2)^3 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^3))^2 in powers of q.
Euler transform of period 16 sequence [ 4, -2, 4, -2, 4, -2, 4, 4, 4, -2, 4, -2, 4, -2, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A215346.
a(n) = -(-1)^n * A215349(n). a(2*n) = 4 * A107035(n). Convolution inverse of A215346.
a(n) ~ exp(sqrt(n)*Pi) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
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EXAMPLE
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q + 4*q^2 + 8*q^3 + 16*q^4 + 30*q^5 + 48*q^6 + 80*q^7 + 128*q^8 + 197*q^9 + ...
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[((1+x^k)^3 * (1-x^k) * (1+x^(8*k))^2 / (1-x^(8*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]*EllipticTheta[2, 0, q^4]/(2*EllipticTheta[3, 0, -q]*EllipticTheta[3, 0, q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 07 2017 *)
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^3))^2, n))}
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CROSSREFS
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Cf. A107035, A215346, A215349.
Sequence in context: A301149 A301143 A215349 * A301144 A298803 A320947
Adjacent sequences: A215345 A215346 A215347 * A215349 A215350 A215351
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Aug 08 2012
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STATUS
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approved
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