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A093831
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Expansion of q * (chi(-q) * chi(-q^5))^-4 in powers of q where chi() is a Ramanujan theta function.
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3
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1, 4, 10, 24, 51, 104, 206, 384, 697, 1228, 2112, 3568, 5898, 9592, 15358, 24256, 37850, 58340, 88980, 134344, 200972, 298112, 438538, 640256, 928041, 1336104, 1911436, 2717776, 3842110, 5401784, 7555012, 10514176, 14562432, 20077672
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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 (eta(q^2) * eta(q^10) / (eta(q) * eta(q^5)))^4 in powers of q.
Euler transform of period 10 sequence [ 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 8*u + 16*u*v).
G.f.: x * (Product_{k>0} (1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^-4.
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (16 * 2^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
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EXAMPLE
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G.f. = q + 4*q^2 + 10*q^3 + 24*q^4 + 51*q^5 + 104*q^6 + 206*q^7 + 384*q^8 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5] )^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[1/((1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, n--; polcoeff( (1 / prod(k=1, (n+5)\10, 1 - x^(10*k - 5), 1 + x * O(x^n)) / prod(k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n)))^4, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^5 + A)))^4, n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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