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A255175
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Expansion of phi(-x) / (1 - x)^2 in powers of x where phi() is a Ramanujan theta function.
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3
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1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: Product_{k>0} (1 - x^(2*k)) * (1 - x^(2*k+1))^2.
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EXAMPLE
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G.f. = 1 - x^2 - 2*x^3 - x^4 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + x^10 - x^12 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / (1 - x)^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^(Mod[k, 2] + 1), {k, 2, n}], {x, 0, n}];
a[ n_] := If[ n < 0, 0, With[{m = Floor[ Sqrt[ n + 1]]}, (-1)^m (n + 1 - m - m^2)]];
Table[Sum[(-1)^(Floor[Sqrt[i]]), {i, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 22 2016 *)
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PROG
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(PARI) {a(n) = my(m); if( n<0, 0, m = sqrtint(n + 1); (-1)^m * (n + 1 - m - m^2))};
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=2, n, (1 - x^k)^(k%2+1), 1 + x * O(x^n)), n))};
(Python)
from math import isqrt
def A255175(n): return ((1+(t:=isqrt(n)))*t-n-1)*(1 if t&1 else -1) # Chai Wah Wu, Aug 04 2022
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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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