A202205 - OEIS

%I #13 Jun 02 2024 20:31:17

%S 1,1,-1,0,-2,2,-1,1,2,-1,-1,2,-1,-4,3,-1,0,-1,0,3,1,-2,0,1,3,-2,-3,2,

%T -2,1,-1,2,-4,-2,5,-1,1,1,-4,4,-1,2,1,-2,2,-1,-1,-3,3,2,-1,-4,0,0,-1,

%U 2,-1,0,2,-1,0,-3,5,-2,-1,5,3,-4,-2,2,-4,4,0,3,-2

%N G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1-x^k)^2.

%C Compare g.f. to: (1 - eta(x))/x = Sum_{n>=0} x^n*Product_{k=1..n} (1-x^k) = 1 + x - x^4 - x^6 + x^11 + x^14 - x^21 - x^25 + x^34 + x^39 +..., where eta(q) is the Dedekind eta function without the q^(1/24) factor.

%H Paul D. Hanna, <a href="/A202205/b202205.txt">Table of n, a(n) for n = 0..1000</a>

%e G.f.: A(x) = 1 + x - x^2 - 2*x^4 + 2*x^5 - x^6 + x^7 + 2*x^8 - x^9 - x^10 +...

%e where A(x) = 1 + x*(1-x)^2 + x^2*(1-x)^2*(1-x^2)^2 + x^3*(1-x)^2*(1-x^2)^2*(1-x^3)^2 +...

%t Join[{1},Rest[CoefficientList[Series[Sum[x^n Product[(1-x^k)^2,{k,n}],{n,100}],{x,0,100}],x]]] (* _Harvey P. Dale_, Jun 02 2024 *)

%o (PARI) {a(n)=polcoeff(1+sum(m=1, n, x^m*prod(k=1, m, (1-x^k +x*O(x^n))^2)), n)}

%Y Cf. A202204.

%K sign

%O 0,5

%A _Paul D. Hanna_, Dec 14 2011