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A047640 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.

%I A047640 #20 Sep 07 2023 02:22:22
%S A047640 1,-15,105,-455,1350,-2793,3625,-765,-9840,29120,-48657,47370,1680,
%T A047640 -111060,252555,-343526,267540,63210,-623510,1216425,-1495173,1093210,
%U A047640 166425,-2073645,3963260,-4864839,3872295,-618310,-4345470,9477960,-12611991
%N A047640 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.
%H A047640 Alois P. Heinz, <a href="/A047640/b047640.txt">Table of n, a(n) for n = 15..10000</a>
%H A047640 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 A047640 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A047640 a(n) = [x^n]( QPochhammer(-x) - 1 )^15. - _G. C. Greubel_, Sep 07 2023
%p A047640 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047640       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047640     end:
%p A047640 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047640       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047640     end:
%p A047640 a:= n-> b(n, 15):
%p A047640 seq(a(n), n=15..45);  # _Alois P. Heinz_, Feb 07 2021
%t A047640 nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^15, {x, 0, nmax}], x]//Drop[#, 15] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047640 With[{k=15}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o A047640 (Magma)
%o A047640 m:=80;
%o A047640 R<x>:=PowerSeriesRing(Integers(), m);
%o A047640 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(15) )); // _G. C. Greubel_, Sep 07 2023
%o A047640 (SageMath)
%o A047640 from sage.modular.etaproducts import qexp_eta
%o A047640 m=75; k=15;
%o A047640 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047640 def A047640_list(prec):
%o A047640     P.<x> = PowerSeriesRing(QQ, prec)
%o A047640     return P( f(k,x) ).list()
%o A047640 a=A047640_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o A047640 (PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^15) \\ _Joerg Arndt_, Sep 07 2023
%Y A047640 Cf. A001482 - A001488, A001490, A047638, A047639, A047641 - A047649, A047654, A047655, A341243.
%K A047640 sign
%O A047640 15,2
%A A047640 _N. J. A. Sloane_
%E A047640 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021