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A199918
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Expansion of false theta series variation of Euler's pentagonal number series in powers of x.
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3
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1, 1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 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, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = b(24*n + 1) where b(n) is multiplicative with b(p^(2*e)) = (-1)^e if p == 13, 17, 29, 23 (mod 24), b(p^(2*e)) = +1 if p = 1, 5, 7, 11 (mod 24) and b(p^(2*e - 1)) = b(2^e) = b(3^e) = 0 if e > 0.
G.f.: 1 + Sum_{k>0} x^k / Product_{i=1..k} (1 + x^(2*i)) = 1 + Sum_{k>0} x^k * Product_{i=1..k-1} (1 + (-x)^i) = Sum_{k in Z} x^((k^2 - 1) / 24) * Kronecker(-24, k).
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EXAMPLE
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G.f. = 1 + x + x^2 + x^5 - x^7 - x^12 - x^15 - x^22 + x^26 + x^35 + x^40 + ...
G.f. = q + q^25 + q^49 + q^121 - q^169 - q^289 - q^361 - q^529 + q^625 + q^841 + ...
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MATHEMATICA
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a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, KroneckerSymbol[ -6, Sqrt[ 24 n + 1]], 0];
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PROG
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(PARI) {a(n) = my(m); if( issquare( 24*n + 1, &m), kronecker( -6, m), 0)};
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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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