A284313 - OEIS

%I #18 Nov 28 2020 19:58:21

%S 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,

%T -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,

%U -14,11,-4,7,-16,15,-6,8,-19,18,-8,9,-21,23,-11,10,-24

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

%H Robert Israel, <a href="/A284313/b284313.txt">Table of n, a(n) for n = 0..10000</a>

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

%F 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

%p V:= Vector(100):

%p V[1]:= 1:

%p for k from 0 to 24 do

%p   V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]

%p od:

%p convert(V,list); # _Robert Israel_, May 03 2017

%t CoefficientList[Series[Product[1 - x^(4k + 1), {k, 0, 100}], {x, 0, 100}], x] (* _Indranil Ghosh_, Mar 25 2017 *)

%o (PARI) Vec(prod(k=0, 100, 1 - x^(4*k + 1)) + O(x^101)) \\ _Indranil Ghosh_, Mar 25 2017

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

%Y Cf. A050449, A169975, A284316.

%K sign,easy

%O 0,15

%A _Seiichi Manyama_, Mar 24 2017