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A131963
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Expansion of f(x, x^2) * f(x^4, x^12) in powers of x where f(, ) is Ramanujan's general theta function.
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11
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1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 3, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 2, 2, 0, 1, 1, 2
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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 psi(x^4) * phi(-x^3) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-13/24) * eta(q^2) * eta(q^3)^2 * eta(q^8)^2 / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, -1, 1, 1, -1, 1, -1, -1, 0, 1, 0, 1, 0, -1, -1, 1, -1, 1, 1, -1, 0, 1, -2, ...].
a(25*n + 13) = a(n). a(25*n + 3) = a(25*n + 8) = a(25*n + 18) = a(25*n + 23) = 0.
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EXAMPLE
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G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + 2*x^12 + x^13 + ...
G.f. = q^13 + q^37 + q^61 + q^109 + 2*q^133 + q^157 + q^181 + q^229 + q^277 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 13}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 04 2015 *)
a[ n_] := SeriesCoefficient[(1/2) x^(-1/2) EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] EllipticTheta[ 2, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 04 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 13; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + 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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