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A128616
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Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function.
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2
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1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0
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OFFSET
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1,19
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COMMENTS
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(iv).
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LINKS
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FORMULA
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Expansion of (eta(q^6) * eta(q^10))^2 / (eta(q^3) * eta(q^5)) in powers of q.
Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].
For n>0, n in A028957 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = A121362(n).
G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(6*k))^2 * (1 + x^(10*k))^2.
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EXAMPLE
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G.f. = x + x^4 + x^6 + x^9 + x^10 + x^15 + x^16 + 2*x^19 + x^24 + x^25 + 2*x^31 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] + KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q(QPochhammer[ q^6] QPochhammer[ q^10])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) + kronecker(20, d) * kronecker(-3, n/d) )/2)};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^2 / (eta(x^3 + A) * eta(x^5 + A)), 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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