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A001487
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^9 in powers of x.
(Formerly M4618 N1971)
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6
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1, -9, 36, -84, 117, -54, -177, 540, -837, 755, -54, -1197, 2535, -3204, 2520, -246, -3150, 6426, -8106, 7011, -2844, -3549, 10359, -15120, 15804, -11403, 2574, 8610, -18972, 25425, -25824, 18954, -6165, -10080, 25101, -35262, 37799, -31374, 17379, 1929
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OFFSET
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9,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^9. - G. C. Greubel, Sep 04 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, 9):
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MATHEMATICA
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nmax=48; CoefficientList[Series[(Product[(1 - (-x)^j), {j, nmax}] -1)^9, {x, 0, nmax}], x]//Drop[#, 9] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^9, {x, 0, 102}], x], 9] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^9 )); // G. C. Greubel, Sep 04 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=9;
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=55, x='x+O('x^N)); Vec((eta(-x)-1)^9) \\ Joerg Arndt, Sep 05 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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