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A128580
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Expansion of phi(x^3) * psi(x^4) - x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
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16
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1, -1, -2, 2, 1, -2, 0, 2, 0, 0, -2, 0, 3, -1, -2, 2, 2, -4, 0, 0, 0, 0, -2, 0, 3, 0, -2, 4, 0, -2, 0, 2, 0, 0, 0, 0, 2, -3, -4, 2, 1, -2, 0, 2, 0, 0, -2, 0, 2, -2, -2, 2, 4, -2, 0, 0, 0, 0, 0, 0, 3, 0, -4, 2, 0, -2, 0, 2, 0, 0, 0, 0, 4, -3, -2, 2, 0, -4, 0, 2, 0, 0, -4, 0, 1, 0, -2, 6, 2, -2, 0, 0, 0, 0, -2, 0, 2, 0, -2, 2, 0, -4, 0, 0, 0
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1)* (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, -2, 0, 0, -1, -1, -1, -1, 0, -2, -1, -2, -1, -2, 0, -1, -1, -1, -1, 0, 0, -2, -1, -2, ...].
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11)= 0.
G.f.: Product_{k>0} (1 - x^(8*k)) * (1 - x^(12*k))^2 / ((1 + x^k) * (1 + x^(2*k))^2 * (1 - x^(3*k)) * (1 + x^(12*k))).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (x^k - x^(3*k)) / (1 + x^(4*k)) * Kronecker(-12, k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) * Kronecker(2, k).
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EXAMPLE
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G.f. = 1 - x - 2*x^2 + 2*x^3 + x^4 - 2*x^5 + 2*x^7 - 2*x^10 + 3*x^12 + ...
G.f. = q - q^3 - 2*q^5 + 2*q^7 + q^9 - 2*q^11 + 2*q^15 - 2*q^21 + 3*q^25 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 12 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker(-12, d) * kronecker(2, n/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^24 + A)), n))};
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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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