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A123863
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Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.
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3
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1, -1, 0, -1, 0, 0, 2, -1, 0, 0, 0, 0, 2, -2, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 1, -2, 0, -2, 0, 0, 2, -1, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -1, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 2, -2, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2, 0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, -3, 0, -1, 0, 0, 2, -2, 0
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OFFSET
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1,7
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COMMENTS
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LINKS
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FORMULA
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Expansion of (a(x) - a(x^2) - a(x^3) - 2*a(x^4) + a(x^6) + 2*a(x^12)) / 6 in powers of x where a() is a cubic AGM theta function. - Michael Somos, Aug 03 2015
Expansion of psi(-x) * psi(-x^9) * phi(x^9) / f(-x^6) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 03 2015
Expansion of eta(q) * eta(q^4) * eta(q^18)^4 / (eta(q^2) * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 1, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = a(6*n + 5) = 0.
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EXAMPLE
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G.f. = q - q^2 - q^4 + 2*q^7 - q^8 + 2*q^13 - 2*q^14 - q^16 + 2*q^19 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {1, -1, 0}[[Mod[#, 3, 1]]], Mod[#, 6] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 03 2015 *)
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 2, 0, x^(9/2)] EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 4, 0, x^18] / (2^(3/2) x^(5/4) QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 03 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^4 / (eta(x^2 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 0, p%6==1, e+1, !(e%2))))};
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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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