A001487 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^9 in powers of x. (Formerly M4618 N1971)

1, -9, 36, -84, 117, -54, -177, 540, -837, 755, -54, -1197, 2535, -3204, 2520, -246, -3150, 6426, -8106, 7011, -2844, -3549, 10359, -15120, 15804, -11403, 2574, 8610, -18972, 25425, -25824, 18954, -6165, -10080, 25101, -35262, 37799, -31374, 17379, 1929

Offset

9,2

References

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

Alois P. Heinz, Table of n, a(n) for n = 9..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 )^9. - G. C. Greubel, Sep 04 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, 9):
seq(a(n), n=9..48);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=48; CoefficientList[Series[(Product[(1 - (-x)^j), {j, nmax}] -1)^9, {x, 0, nmax}], x]//Drop[#, 9] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^9, {x, 0, 102}], x], 9] (* 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)^9 )); // G. C. Greubel, Sep 04 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=9;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001487_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A001487_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^9) \\ Joerg Arndt, Sep 05 2023

Crossrefs

Cf. A001482 - A001486, A001488, A047638 - A047649, A047654, A047655, A341243.

Sequence in context: A083353 A083014 A202287 * A175673 A285198 A230212

Adjacent sequences: A001484 A001485 A001486 * A001488 A001489 A001490

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved