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A260114
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Expansion of f(x)^4 * phi(-x^3) / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.
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1
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1, 6, 14, 18, 21, 30, 38, 42, 43, 48, 62, 66, 74, 78, 64, 84, 98, 102, 110, 96, 133, 126, 108, 138, 112, 150, 158, 162, 183, 126, 182, 192, 194, 198, 160, 210, 180, 222, 230, 192, 242, 252, 288, 228, 208, 270, 278, 282, 273, 240, 252, 306, 314, 336, 294, 330
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/6) * eta(q^2)^13 * eta(q^3)^2 / (eta(q)^6 * eta(q^4)^4 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 6, -7, 4, -3, 6, -8, 6, -3, 4, -7, 6, -4, ...].
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EXAMPLE
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G.f. = 1 + 6*x + 14*x^2 + 18*x^3 + 21*x^4 + 30*x^5 + 38*x^6 + 42*x^7 + ...
G.f. = q + 6*q^7 + 14*q^13 + 18*q^19 + 21*q^25 + 30*q^31 + 38*q^37 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, # KroneckerSymbol[ -3, #] KroneckerSymbol[ -4, m/#] &]]];
a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #] &]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4 EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x], {x, 0, n}];
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PROG
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(PARI) {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, d * kronecker( -3, d) * kronecker( -4, m/d)))};
(PARI) {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, m/d * kronecker( 12, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 * eta(x^3 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^4 * 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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