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A193275
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Expansion of psi(x^2) * phi(x^17) - x^4 * phi(x) * psi(x^34) in powers of x^4 where phi(), psi() are Ramanujan theta functions.
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1
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1, 0, 1, 0, -1, -2, 1, 0, -2, 0, 0, 0, 1, -2, 0, 0, 0, 2, 0, 2, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, -1, -2, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 1, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, -2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0
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OFFSET
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0,6
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COMMENTS
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LINKS
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FORMULA
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Expansion of (phi(q) * phi(q^68) - phi(q^4) * phi(q^17)) / (2 * q) in powers of q^4 where phi() is a Ramanujan theta function.
Expansion of q^(-1) * (eta(q^4)^2 * eta(q^34)^5 / (eta(q^2) * eta(q^17)^2 * eta(q^68)^2) - eta(q^2)^5 * eta(q^68)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^34))) in powers of q^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (272 t)) = 272^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
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EXAMPLE
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1 + x^2 - x^4 - 2*x^5 + x^6 - 2*x^8 + x^12 - 2*x^13 + 2*x^17 + 2*x^19 - ...
q + q^9 - q^17 - 2*q^21 + q^25 - 2*q^33 + q^49 - 2*q^53 + 2*q^69 + 2*q^77 - ...
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MATHEMATICA
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a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q^(1/4)]*EllipticTheta[3, 0, q^17] - EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^(17/4)])/(2* q^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 03 2018 *)
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PROG
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(PARI) {a(n) = local(x); if( n<0, 0, n = 4*n + 1; (sum( i=1, sqrtint( n\68), issquare( n - 68*i^2)) - sum( i=1, sqrtint( (n-1)\17), issquare( n - 17*i^2, &x) && (x%2==0) )) * 2 + issquare( n) - issquare( 17*n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x^2 + A) * eta(x^17 + A)^2 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^34 + 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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