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A204386
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Expansion of (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.
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1
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1, 12, 28, 96, 126, 336, 344, 768, 757, 1512, 1332, 2688, 2198, 4128, 3528, 6144, 4914, 9084, 6860, 12096, 9632, 15984, 12168, 21504, 15751, 26376, 20440, 33024, 24390, 42336, 29792, 49152, 37296, 58968, 43344, 72672, 50654, 82320, 61544, 96768
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of x * psi(x)^8 + 4 * x^2 * psi(x^2)^8 in powers of x where psi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / eta(q))^8 + 4 * (eta(q^4)^2 / eta(q^2))^8 in powers of q.
a(n) is multiplicative with a(2^e) = 3/2 * 8^e if e>0, a(p^e) = ((p^3) ^ (e+1) - 1) / (p^3 - 1).
Dirichlet g.f.: (1 + 1/2^(s-2)) * (1 - 1/2^s) * zeta(s-3) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 5*Pi^4/1536 = 0.317086... . (End)
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EXAMPLE
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x + 12*x^2 + 28*x^3 + 96*x^4 + 126*x^5 + 336*x^6 + 344*x^7 + 768*x^8 + ...
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MATHEMATICA
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a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^(1/2)]^8 + 4*EllipticTheta[2, 0, q]^8)/256, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Apr 13 2018 *)
CoefficientList[Series[(EllipticTheta[2, 0, q^(1/2)]^8 +4*EllipticTheta[2, 0, q]^8)/ 256, {q, 0, 50}], q] (* Vaclav Kotesovec, Apr 13 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, if( n%2, sigma( n, 3), 12 * sumdiv( n/2, d, (n/2/d%2) * d^3)))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8 + 4 * x * (eta(x^4 + A)^2 / eta(x^2 + A))^8, n))}
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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