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A213624
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Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.
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10
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1, 2, 1, 2, 3, 2, 4, 4, 2, 2, 5, 4, 2, 6, 3, 6, 7, 2, 5, 4, 5, 6, 6, 2, 5, 10, 3, 6, 10, 4, 6, 8, 3, 8, 7, 6, 7, 6, 4, 6, 11, 6, 9, 10, 3, 6, 14, 4, 8, 10, 8, 10, 5, 6, 4, 16, 7, 4, 10, 4, 13, 14, 7, 8, 8, 6, 10, 12, 7, 12, 15, 8, 8, 10, 4, 6, 17, 6, 10, 10
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OFFSET
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0,2
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COMMENTS
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Dickson writes that Liouville proved several related results about sums of triangular number. In particular, that every nonnegative integer is the sum of t1 + t2 + 4*t3 where t1, t2, t3 are trianglular numbers. - Michael Somos, Feb 10 2020
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 23.
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LINKS
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FORMULA
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Expansion of q^(-3/4) * eta(q^2)^4 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].
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EXAMPLE
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G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q^3 + 2*q^7 + q^11 + 2*q^15 + 3*q^19 + 2*q^23 + 4*q^27 + 4*q^31 + 2*q^35 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];
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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^2 + A)^4 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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