|
%I M4845 N2071 #21 Sep 06 2023 01:31:08
%S 1,-12,66,-220,483,-660,252,1320,-4059,6644,-6336,240,12255,-27192,
%T 35850,-27972,-2343,50568,-99286,122496,-96162,11584,115116,-242616,
%U 315216,-283800,128304,126280,-409398,622644,-671550,501468,-122508,-382360
%N Expansion of {Product_{j>=1} (1 - (-x)^j) - 1}^12 in powers of x.
%D H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%D M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A001490/b001490.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: (eta(z)*eta(6*z)/(eta(2*z)*eta(3*z)))^12.
%F a(n) = [x^n]( QPochhammer(-x) - 1 )^12. - _G. C. Greubel_, Sep 05 2023
%t With[{k=12}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 102}], x], k]] (* _G. C. Greubel_, Sep 04 2023 *)
%o (Magma)
%o m:=102;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(12) )); // _G. C. Greubel_, Sep 05 2023
%o (SageMath)
%o from sage.modular.etaproducts import qexp_eta
%o m=100; k=12;
%o def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o def A001490_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(k,x) ).list()
%o a=A001490_list(m); a[k:] # _G. C. Greubel_, Sep 05 2023
%o (PARI) my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^12) \\ _Joerg Arndt_, Sep 05 2023
%Y Cf. A001482 - A001488, A047638 - A047649, A047654, A047655, A341243.
%K sign
%O 1,2
%A _N. J. A. Sloane_
|
