A001482 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x. (Formerly M3263 N1317)

1, -4, 6, -4, -3, 12, -16, 16, -6, -8, 18, -28, 26, -20, 2, 12, -23, 32, -36, 28, -6, 4, 22, -20, 39, -32, 32, -12, 2, 16, -12, 24, -40, 28, -34, 0, -6, -16, 0, -40, 6, -36, 26, -32, -5, 0, -20, 8, -16, 12, -10, 40, -22, 12, 14, 12, 45, 16, 38, 4, 12, 0, 34, 8, 38, 12, -24, 44, 2, 16

Offset

4,2

References

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

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 = 4..10000

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

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, 4):
seq(a(n), n=4..73);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 73; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(1 -QPochhammer[-x])^4, {x, 0, 100}], x], 4] (* G. C. Greubel, Sep 04 2023 *)

Prog

(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (1 - (&*[1-(-x)^j: j in [1..m+2]]))^4 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100
def f4(x): return (1 - product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1, m+2) ) )^4
def A001482_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f4(x) ).list()
a=A001482_list(m); a[4:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ Joerg Arndt, Sep 04 2023

Crossrefs

Cf. A001483 - A001488, A047638 - A047649, A047654, A047655, A341243.

Sequence in context: A349942 A187147 A128633 * A198493 A161965 A256419

Adjacent sequences: A001479 A001480 A001481 * A001483 A001484 A001485

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved