A047639 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^14 in powers of x.

1, -14, 91, -364, 987, -1820, 1897, 754, -8008, 18928, -26845, 19460, 15015, -76272, 141065, -163072, 90727, 99386, -368277, 602616, -643734, 358190, 274547, -1101100, 1801086, -1982330, 1344525, 148316, -2163590, 4032756, -4938843, 4216576

Offset

14,2

Links

Alois P. Heinz, Table of n, a(n) for n = 14..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)

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^14. - G. C. Greubel, Sep 07 2023

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 14):
seq(a(n), n=14..45);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^14, {x, 0, nmax}], x]//Drop[#, 14] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=14}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

Prog

(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(14) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=14;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047639_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047639_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^14) \\ Joerg Arndt, Sep 07 2023

Crossrefs

Cf. A001482 - A001488, A001490, A047638, A047640 - A047649, A047654, A047655, A341243.

Sequence in context: A200191 A266805 A301380 * A202291 A010930 A220892

Adjacent sequences: A047636 A047637 A047638 * A047640 A047641 A047642

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved