A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 3, -2, 0, -1, 3, -3, 1, -1, 4, -4, 1, -1, 4, -5, 2, -1, 5, -7, 3, -1, 5, -8, 5, -2, 6, -10, 6, -2, 6, -12, 9, -3, 7, -14, 11, -4, 7, -16, 15, -6, 8, -19, 18, -8, 9, -21, 23, -11, 10, -24

Offset: 0

Author

Seiichi Manyama, Mar 24 2017

Keywords

• sign
• easy

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), this sequence (m=4), A284314 (m=5).

Cf. A050449, A169975, A284316.

Programs

Maple

V:= Vector(100):
V[1]:= 1:
for k from 0 to 24 do
  V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]
od:
convert(V,list); # Robert Israel, May 03 2017

Mathematica

CoefficientList[Series[Product[1 - x^(4k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

PARI

Vec(prod(k=0, 100, 1 - x^(4*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n-1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284316. - Peter Bala, Nov 28 2020