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A214341
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Expansion of 1 / k(q) = 1 / (r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.
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2
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1, 1, 2, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 4, 1, -2, -6, -8, -7, -3, 4, 10, 14, 12, 6, -6, -16, -22, -20, -8, 8, 26, 34, 31, 12, -14, -41, -54, -47, -20, 23, 61, 84, 72, 31, -32, -90, -122, -107, -44, 45, 133, 174, 154, 61, -68, -192, -254, -220, -90, 100
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OFFSET
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-1,3
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COMMENTS
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Number 12 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(10). [Yang 2004] - Michael Somos, Aug 07 2014
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LINKS
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FORMULA
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Expansion of (1/x) * (f(-x^4, -x^6) * f(-x^3, -x^7)) / (f(-x^2, -x^8) * f(-x, -x^9)) in powers of x where f(,) is Ramanujan's two-variable theta function.
Euler transform of period 10 sequence [ 1, 1, -1, -1, 0, -1, -1, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (u^2 - 1).
G.f.: (1/x) * Product_{k>0} (1 - x^(10*k - 3)) * (1 - x^(10*k - 4)) * (1 - x^(10*k - 6)) * (1 - x^(10*k - 7)) /((1 - x^(10*k - 1)) * (1 - x^(10*k - 2)) * (1 - x^(10*k - 8)) * (1 - x^(10*k - 9))).
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EXAMPLE
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G.f. = 1/q + 1 + 2*q + q^2 + q^3 - q^5 - 2*q^6 - 2*q^7 - q^8 + q^9 + 3*q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/q Product[(1 - q^k)^{-1, -1, 1, 1, 0, 1, 1, -1, -1, 0}[[Mod[k, 10, 1]]], {k, n + 1}], {q, 0, n}]; (* Michael Somos, Aug 07 2014 *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, -1, -1, 1, 1, 0, 1, 1, -1, -1][k%10 + 1]), 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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