A047645 - OEIS

%I #21 Sep 06 2023 01:35:03

%S 1,-20,190,-1140,4825,-15124,35320,-57760,45220,80560,-405954,910460,

%T -1289340,852340,1259530,-5357924,10151510,-12048660,5883350,12186960,

%U -40135713,66244280,-69648870,28191460,66920755,-195366168,300881530

%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^20 in powers of x.

%H Alois P. Heinz, <a href="/A047645/b047645.txt">Table of n, a(n) for n = 20..10000</a>

%H H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)

%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.

%F a(n) = [x^n]( QPochhammer(-x) - 1 )^20. - _G. C. Greubel_, Sep 06 2023

%p g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]

%p       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)

%p     end:

%p b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),

%p       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p     end:

%p a:= n-> b(n, 20):

%p seq(a(n), n=20..46);  # _Alois P. Heinz_, Feb 07 2021

%t nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^20, {x,0,nmax}], x]//Drop[#, 20] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)

%t With[{k=20}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 06 2023 *)

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(20) )); // _G. C. Greubel_, Sep 06 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=75; k=20;

%o def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k

%o def A047645_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( f(k,x) ).list()

%o a=A047645_list(m); a[k:] # _G. C. Greubel_, Sep 06 2023

%o (PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^20) \\ _Joerg Arndt_, Sep 06 2023

%Y Cf. A001482 - A001488, A001490, A047638 - A047644, A047646 - A047649, A047654, A047655, A341243.

%K sign

%O 20,2

%A _N. J. A. Sloane_

%E Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021