A047642 - OEIS

%I #18 Sep 07 2023 02:22:14

%S 1,-17,136,-680,2363,-5916,10319,-9656,-8534,57426,-133076,190383,

%T -134810,-140148,657611,-1240116,1461337,-770917,-1171504,4061946,

%U -6678161,7071269,-3376863,-4939180,15963612,-25098443,26265408,-14513461,-10810368,43792034

%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^17 in powers of x.

%H Alois P. Heinz, <a href="/A047642/b047642.txt">Table of n, a(n) for n = 17..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 )^17. - _G. C. Greubel_, Sep 07 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, 17):

%p seq(a(n), n=17..46);  # _Alois P. Heinz_, Feb 07 2021

%t nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^17, {x, 0, nmax}], x]//Drop[#, 17] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)

%t With[{k=17}, 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)^(17) )); // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=75; k=17;

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

%o def A047642_list(prec):

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

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

%o a=A047642_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023

%o (PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^17) \\ _Joerg Arndt_, Sep 07 2023

%Y Cf. A001482 - A001488, A001490, A047638 - A047641, A047643 - A047649, A047654, A047655, A341243.

%K sign

%O 17,2

%A _N. J. A. Sloane_

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