A047638 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^13 in powers of x.

1, -13, 78, -286, 702, -1131, 845, 1300, -5928, 11583, -13715, 5915, 15834, -47477, 73658, -71201, 20436, 79391, -198796, 280345, -258557, 92807, 200850, -536341, 773916, -768222, 432705, 204477, -979628, 1626196, -1856569, 1471184, -452192

Offset: 13

Author

N. J. A. Sloane

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 13..10000
• H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
• H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Crossrefs

Cf. A001482 - A001488, A001490, A047639 - A047649, A047654, A047655, A341243.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(13) )); // G. C. Greubel, Sep 07 2023

Maple

N:= 100: # to get a(13)..a(N)
G:= (mul(1-(-x)^j,j=1..N)-1)^13:
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=13..N); # Robert Israel, Aug 08 2018

Mathematica

With[{k=13}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

PARI

my(x='x+O('x^40)); Vec((eta(-x)-1)^13) \\ Joerg Arndt, Sep 07 2023

SageMath

from sage.modular.etaproducts import qexp_eta
m=75; k=13;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047638_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047638_list(m); a[k:] # G. C. Greubel, Sep 07 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^13. - G. C. Greubel, Sep 07 2023

Extensions

Definition corrected by Robert Israel, Aug 08 2018