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A213791
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Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.
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2
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1, -6, 15, -26, 45, -66, 82, -120, 156, -170, 231, -276, 290, -390, 435, -438, 561, -630, 651, -780, 861, -842, 1020, -1170, 1095, -1326, 1431, -1370, 1716, -1740, 1682, -2016, 2145, -2132, 2415, -2550, 2353, -2850, 3120, -2810, 3321, -3486, 3285, -3906, 4005
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. X
J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.176
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LINKS
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FORMULA
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Expansion of q^(-3/4) * ( eta(q) * eta(q^4) / eta(q^2) )^6 in powers of q.
Expansion of -1/(8 * r) * ( 1^2 * r^1 / (1 + q) - 3^2 * q^(3/4) / (1 + q^3) - 5^2 * r^5 / (1 + q^5) + 7^2 * q^(7/4) / (1 + q^7) + 9^2 * r^9 / (1 + q^9) - ...) in powers of q where r = q^(3/4) [Glaisher 1876].
Expansion of q^(-1/4) * ( sqrt(k * k') * K / Pi )^3 in powers of q where k, k', K are Jacobi elliptic functions. [Jacobi 1828, p. 108 quoted in Glaisher 1876, p. 176].
Euler transform of period 4 sequence [ -6, 0, -6, -6, ...].
G.f.: (Sum_{k>0} (-x)^((k^2 - k)/2))^6.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 45*x^4 - 66*x^5 + 82*x^6 - 120*x^7 + ...
G.f. = q^3 - 6*q^7 + 15*q^11 - 26*q^15 + 45*q^19 - 66*q^23 + 82*q^27 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^2, x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^6, 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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