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%I #25 Sep 07 2023 02:21:58
%S 1,-13,78,-286,702,-1131,845,1300,-5928,11583,-13715,5915,15834,
%T -47477,73658,-71201,20436,79391,-198796,280345,-258557,92807,200850,
%U -536341,773916,-768222,432705,204477,-979628,1626196,-1856569,1471184,-452192
%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^13 in powers of x.
%H Robert Israel, <a href="/A047638/b047638.txt">Table of n, a(n) for n = 13..10000</a>
%H H. Gupta, <a href="https://doi.org/10.1112/jlms/s1-39.1.433">On the coefficients of the powers of Dedekind's modular form</a>, J. London Math. Soc., 39 (1964), 433-440.
%H H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F a(n) = [x^n]( QPochhammer(-x) - 1 )^13. - _G. C. Greubel_, Sep 07 2023
%p N:= 100: # to get a(13)..a(N)
%p G:= (mul(1-(-x)^j,j=1..N)-1)^13:
%p S:= series(G,x,N+1):
%p seq(coeff(S,x,n),n=13..N); # _Robert Israel_, Aug 08 2018
%t With[{k=13}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o (Magma)
%o m:=80;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(13) )); // _G. C. Greubel_, Sep 07 2023
%o (SageMath)
%o from sage.modular.etaproducts import qexp_eta
%o m=75; k=13;
%o def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o def A047638_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(k,x) ).list()
%o a=A047638_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o (PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^13) \\ _Joerg Arndt_, Sep 07 2023
%Y Cf. A001482 - A001488, A001490, A047639 - A047649, A047654, A047655, A341243.
%K sign
%O 13,2
%A _N. J. A. Sloane_
%E Definition corrected by _Robert Israel_, Aug 08 2018
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