A001483 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x. (Formerly M3791 N1546)

1, -5, 10, -10, 0, 19, -35, 40, -25, -10, 45, -75, 80, -60, 15, 45, -85, 115, -115, 90, -21, -35, 95, -130, 135, -135, 70, -35, -65, 105, -146, 120, -150, 90, -65, -25, 90, -115, 150, -125, 130, -45, 80, 35, -5, 160, -110, 170, -85, 95, 25, 50, 0, -60, 95, -116, 120, -135

Offset

5,2

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 = 5..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 )^5. - 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, 5):
seq(a(n), n=5..62);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 62; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^5, {x, 0, 102}], x], 5] (* 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)^5 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100; k=5;
def f(k, x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1, m+2) ) )^k
def A001483_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A001483_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ Joerg Arndt, Sep 04 2023

Crossrefs

Cf. A001482, A001484 - A001488, A047638 - A047649, A047654, A047655, A341243.

Sequence in context: A144136 A343852 A198286 * A173679 A230208 A168228

Adjacent sequences: A001480 A001481 A001482 * A001484 A001485 A001486

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved