A001490 Expansion of {Product_{j>=1} (1 - (-x)^j) - 1}^12 in powers of x. (Formerly M4845 N2071)

1, -12, 66, -220, 483, -660, 252, 1320, -4059, 6644, -6336, 240, 12255, -27192, 35850, -27972, -2343, 50568, -99286, 122496, -96162, 11584, 115116, -242616, 315216, -283800, 128304, 126280, -409398, 622644, -671550, 501468, -122508, -382360

Offset

1,2

References

H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

G.f.: (eta(z)*eta(6*z)/(eta(2*z)*eta(3*z)))^12.

a(n) = [x^n]( QPochhammer(-x) - 1 )^12. - G. C. Greubel, Sep 05 2023

Mathematica

With[{k=12}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 102}], x], k]] (* G. C. Greubel, Sep 04 2023 *)

Prog

(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(12) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=12;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001490_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A001490_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^12) \\ Joerg Arndt, Sep 05 2023

Crossrefs

Cf. A001482 - A001488, A047638 - A047649, A047654, A047655, A341243.

Sequence in context: A289223 A296914 A285580 * A010928 A080559 A284641

Adjacent sequences: A001487 A001488 A001489 * A001491 A001492 A001493

Keyword

sign

Author

N. J. A. Sloane

Status

approved