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A109127
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Expansion of q^(-1/8) * (eta(q^3) - eta(q)^3) / 3 in powers of q.
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0
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1, 0, -2, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: Sum_{k>1} (-1)^k * floor((2*k)/3) * x^((k^2 - k)/2).
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EXAMPLE
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G.f. = x - 2*x^3 + 2*x^6 - 3*x^10 + 4*x^15 - 4*x^21 + 5*x^28 - 6*x^36 + 6*x^45 - ...
G.f. = q^9 - 2*q^25 + 2*q^49 - 3*q^81 + 4*q^121 - 4*q^169 + 5*q^225 - 6*q^289 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] - QPochhammer[ x]^3) / 3, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := Which[ n < 0, 0, EvenQ[ Length @ Divisors [8 n + 1]], 0, True, With[ {k = Sqrt[8 n + 1] + 1}, Quotient[k, 3] (-1)^Quotient[k, 2]]]; (* Michael Somos, Apr 26 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); 1/3 * polcoeff( eta(x^3 + A) - eta(x + A)^3, n))};
(PARI) {a(n) = if( n<0, 0, if( issquare(8*n + 1, &n), ((n+1) \ 3) * (-1)^((n+1) / 2)))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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