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A118271
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Expansion of (9 * theta_4(q^3)^4 - theta_4(q)^4) / 8 in powers of q.
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5
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1, 1, -3, -5, -3, 6, 15, 8, -3, -23, -18, 12, 15, 14, -24, -30, -3, 18, 69, 20, -18, -40, -36, 24, 15, 31, -42, -77, -24, 30, 90, 32, -3, -60, -54, 48, 69, 38, -60, -70, -18, 42, 120, 44, -36, -138, -72, 48, 15, 57, -93, -90, -42, 54, 231, 72, -24, -100, -90, 60, 90, 62, -96, -184, -3, 84, 180, 68, -54, -120, -144
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OFFSET
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0,3
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LINKS
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FORMULA
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Expansion of eta(q^2)^5 * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q.
Expansion of b(q^2) * (4*b(q^4) - b(q)) / 3 in powers of q where b() is a cubic AGM theta function.
Euler transform of period 6 sequence [ 1, -4, -2, -4, 1, -4, ...].
a(n) is multiplicative with a(2^e) = -3 if e>0, a(3^e) = 4 - 3^(e+1), a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 2^(2-s)) * (1 - 2^(1-s)) * (1 - 3^(2-s)). - Amiram Eldar, Oct 28 2023
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EXAMPLE
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1 + q - 3*q^2 - 5*q^3 - 3*q^4 + 6*q^5 + 15*q^6 + 8*q^7 - 3*q^8 - ...
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MATHEMATICA
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eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[eta[q^2]^5 *eta[q^3]^3/(eta[q]*eta[q^6]^3), {q, 0, 55}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 11 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * (-x)^k^2, 1 + x * O(x^n))^4 - 9 * sum( k=1, sqrtint(n\3), 2 * (-x^3)^k^2, 1 + x * O(x^n))^4, n) / -8)}
(PARI) {a(n) = if( n<1, n==0, -(-1)^n * ( sumdiv( n, d, d * (1 - if( d%3==0, 3) - if( d%4==0, 1) + if(d%12==0, 3)))))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^3 / eta(x + A) / eta(x^6 + A)^3, 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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