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A113421
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Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^12) / eta(q^3)^2 in powers of q.
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3
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1, -2, -1, 4, -4, 2, 6, -8, 1, 8, -12, -4, 14, -12, 4, 16, -16, -2, 18, -16, -6, 24, -24, 8, 21, -28, -1, 24, -28, -8, 30, -32, 12, 32, -24, 4, 38, -36, -14, 32, -40, 12, 42, -48, -4, 48, -48, -16, 43, -42, 16, 56, -52, 2, 48, -48, -18, 56, -60, 16, 62, -60, 6, 64, -56, -24, 66, -64, 24, 48, -72, -8, 74, -76, -21, 72
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OFFSET
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1,2
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COMMENTS
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Number 26 of the 74 eta-quotients listed in Table I of Martin (1996).
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LINKS
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FORMULA
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Euler transform of period 12 sequence [-2, -2, 0, -3, -2, -2, -2, -3, 0, -2, -2, -4, ...].
G.f.: Sum_{k>0} (3*k - 2) * x^(3*k - 2) / (1 + x^(6*k - 4)) - (3*k - 1) * x^(3*k - 1) / (1 + x^(6*k - 2)).
G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) * (1 - x^(2*k - 1))^2 * (1 - x^(4*k - 2)) / (1 - x^(6*k - 3))^2.
a(n) is multiplicative with a(2^e) = (-2)^e, a(3^e) = (-1)^e, a(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * Kronecker( -3, p) and y = (-1)^[p/2].
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EXAMPLE
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G.f. = q - 2*q^2 - q^3 + 4*q^4 - 4*q^5 + 2*q^6 + 6*q^7 - 8*q^8 + q^9 + 8*q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^4] QPochhammer[ q^6]^2 QPochhammer[ q^12] / QPochhammer[ q^3]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d * kronecker( -3, d) * (-1)^(n / d \ 2)))};
(PARI) {a(n) = my(A, p, e, t); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, p==3, (-1)^e, t = (-1)^(p\2); p *= kronecker( -3, p); (p^(e+1) - t^(e+1)) / (p - t))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / eta(x^3 + A)^2, n))};
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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