A047649 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^11 in powers of x.

1, -11, 55, -165, 319, -352, -44, 1100, -2585, 3542, -2519, -1530, 8085, -14410, 16170, -9460, -6644, 28105, -46145, 50248, -32802, -6193, 57200, -102575, 121968, -100397, 35123, 60390, -158840, 226413, -234344, 168773, -37070, -131175, 290851, -391402

Offset

11,2

Links

Alois P. Heinz, Table of n, a(n) for n = 11..10000

H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

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

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^11. - G. C. Greubel, Sep 05 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, 11):
seq(a(n), n=11..46);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^11, {x, 0, nmax}], x]//Drop[#, 11] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=11}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)

Prog

(Magma)
m:=75;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(11) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=11;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047649_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047649_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^11) \\ Joerg Arndt, Sep 05 2023

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047648, A047654, A047655, A341243.

Sequence in context: A246406 A255415 A153449 * A010927 A009550 A226255

Adjacent sequences: A047646 A047647 A047648 * A047650 A047651 A047652

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved