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A233693
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Expansion of q * psi(-q) * chi(-q^6) * psi(-q^9) / (phi(-q) * phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
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4
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1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 48, 60, 75, 92, 114, 140, 170, 208, 252, 304, 366, 439, 526, 626, 744, 884, 1044, 1232, 1451, 1704, 1998, 2336, 2730, 3182, 3700, 4300, 4986, 5772, 6672, 7700, 8876, 10212, 11736, 13472, 15438, 17673, 20207, 23076
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q^4) * eta(q^6) * eta(q^9) * eta(q^36)^2 / (eta(q) * eta(q^12) * eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, ...].
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
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EXAMPLE
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G.f. = q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 8*q^7 + 11*q^8 + 14*q^9 + ...
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(6*k)) * (1-x^(9*k)) * (1+x^(18*k))^2 / ((1-x^k) * (1-x^(12*k)) * (1-x^(18*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
QP := QPochhammer; A233693[n_]:= SeriesCoefficient[QP[q^4]*QP[q^6] *QP[q^9]*QP[q^36]^2/(QP[q]* QP[q^12]*QP[q^18]^3), {q, 0, n}]; Table[A233693[n], {n, 0, 50}] (* G. C. Greubel, Dec 25 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^4 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)^2 / (eta(x + A) * eta(x^12 + A) * eta(x^18 + A)^3), 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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STATUS
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approved
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