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A113429
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Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.
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8
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1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - Wolfdieter Lang, Oct 28 2016
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REFERENCES
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G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.
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LINKS
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FORMULA
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Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...].
|a(n)| is the characteristic function of A085787.
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2).
f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function.
G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - Joerg Arndt, Apr 07 2011
a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1.
(End)
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1)). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - Wolfdieter Lang, Nov 01 2016
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EXAMPLE
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G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ...
G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)
a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* Michael Somos, Jun 26 2017 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};
(PARI) {a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* Michael Somos, Oct 29 2016 */
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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