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A047645
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^20 in powers of x.
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4
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1, -20, 190, -1140, 4825, -15124, 35320, -57760, 45220, 80560, -405954, 910460, -1289340, 852340, 1259530, -5357924, 10151510, -12048660, 5883350, 12186960, -40135713, 66244280, -69648870, 28191460, 66920755, -195366168, 300881530
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OFFSET
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20,2
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^20. - G. C. Greubel, Sep 06 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, 20):
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MATHEMATICA
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nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^20, {x, 0, nmax}], x]//Drop[#, 20] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=20}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 06 2023 *)
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PROG
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(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(20) )); // G. C. Greubel, Sep 06 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=20;
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()
(PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^20) \\ Joerg Arndt, Sep 06 2023
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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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