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A131962
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Expansion of psi(x) * phi(-x^12) / chi(-x^4) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
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10
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1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 0, 3, 1, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2
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OFFSET
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0,11
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-7/24) * eta(q^2)^2 * eta(q^8) * eta(q^12)^2/( eta(q) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -2, ...].
a(25*n + 7) = a(n). a(25*n + 2) = a(25*n + 12) = a(25*n + 17) = a(25*n + 22) = 0.
Expansion of chi(x) * f(-x^8) * phi(-x^12) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015
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EXAMPLE
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G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^7 + q^31 + q^79 + q^103 + q^127 + q^151 + q^175 + q^199 + q^223 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 7}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x^4, x^4], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 24*n + 7; 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)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + 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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