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A047654
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.
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25
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1, -2, 1, 0, -2, 2, -2, 2, 1, 0, 2, -2, 3, 0, 2, 0, 0, 2, -2, 0, -2, 2, -1, 0, 0, -2, -2, -2, 1, -2, 0, -2, -2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, -2, 2, 2, 0, 2, 0, 2, 0, 2, 2, 0, -4, 0, 0, 2, 1, -2, 0, -2, 0, 0, 0, 0, 2, -4, 1, 0, 0, -2, -2, -2, -2, 0, 0, -2, 0, 2, -2, 2, -2
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OFFSET
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2,2
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - G. C. Greubel, Sep 07 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 2):
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MATHEMATICA
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nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
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PROG
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(PARI) seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ Andrew Howroyd, Feb 07 2021
(Magma)
m:=120;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=125; k=2;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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