cp's OEIS Frontend

A047648 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^23 in powers of x.

%I A047648 #21 Sep 05 2023 12:25:53
%S A047648 1,-23,253,-1771,8832,-33143,95611,-209231,317009,-181401,-686642,
%T A047648 2828977,-6099278,8422623,-4906406,-10919687,41968146,-78977952,
%U A047648 93297545,-40351223,-117265247,367581446,-606562624,631382751,-207879980,-777907725,2132043121
%N A047648 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^23 in powers of x.
%H A047648 Alois P. Heinz, <a href="/A047648/b047648.txt">Table of n, a(n) for n = 23..10000</a>
%H A047648 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%H A047648 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.
%F A047648 a(n) = [x^n]( QPochhammer(-x) - 1 )^23. - _G. C. Greubel_, Sep 05 2023
%p A047648 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047648       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047648     end:
%p A047648 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047648       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047648     end:
%p A047648 a:= n-> b(n, 23):
%p A047648 seq(a(n), n=23..49);  # _Alois P. Heinz_, Feb 07 2021
%t A047648 nmax=49; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^23, {x,0,nmax}], x]//Drop[#, 23] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047648 With[{k=23}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 05 2023 *)
%o A047648 (Magma)
%o A047648 m:=75;
%o A047648 R<x>:=PowerSeriesRing(Integers(), m);
%o A047648 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(23) )); // _G. C. Greubel_, Sep 05 2023
%o A047648 (SageMath)
%o A047648 from sage.modular.etaproducts import qexp_eta
%o A047648 m=75; k=23;
%o A047648 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047648 def A047648_list(prec):
%o A047648     P.<x> = PowerSeriesRing(QQ, prec)
%o A047648     return P( f(k,x) ).list()
%o A047648 a=A047648_list(m); a[k:] # _G. C. Greubel_, Sep 05 2023
%o A047648 (PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^23) \\ _Joerg Arndt_, Sep 05 2023
%Y A047648 Cf. A001482 - A001488, A001490, A047638 - A047647, A047649, A047654, A047655, A341243.
%K A047648 sign
%O A047648 23,2
%A A047648 _N. J. A. Sloane_
%E A047648 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021