A260180 - OEIS

%I #13 May 16 2017 08:51:00

%S 1,1,0,1,-1,1,-1,1,-3,4,-4,1,0,1,-6,11,-11,1,7,1,-18,22,-10,1,-3,6,

%T -12,37,-48,1,45,1,-71,56,-16,36,-41,1,-18,79,-69,1,51,1,-186,232,-22,

%U 1,-179,8,186,137,-311,1,1,331,-364,172,-28,1,-51,1,-30,295,-599,716,-263,1,-713,254,1177,1

%N G.f.: Sum_{n>=0} x^n * (1 - x^n)^n.

%C Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.

%H Paul D. Hanna, <a href="/A260180/b260180.txt">Table of n, a(n) for n = 0..2050</a>

%F G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2-n) / (1 - x^n)^n.

%F G.f.: Sum_{n>=1} - x^(-n) / (1 - x^(-n))^n.

%e G.f.: A(x) = 1 + x + x^3 - x^4 + x^5 - x^6 + x^7 - 3*x^8 + 4*x^9 - 4*x^10 +...

%e where

%e A(x) = 1 + x*(1-x) + x^2*(1-x^2)^2 + x^3*(1-x^3)^3 + x^4*(1-x^4)^4 + x^5*(1-x^5)^5 +...

%e Also,

%e A(x) = 1/(1-x) - x^2/(1-x^2)^2 + x^6/(1-x^3)^3 - x^12/(1-x^4)^4 + x^20/(1-x^5)^5 +...

%t terms = 100; 1 + Sum[x^n*(1 - x^n)^n, {n, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, May 16 2017 *)

%o (PARI) {a(n) = local(A=1); A = sum(k=0, n+1, x^k*(1-x^k)^k + O(x^(n+2))); polcoeff(A, n)}

%o for(n=0, 80, print1(a(n), ", "))

%o (PARI) {a(n) = local(A=1); A = sum(k=1, n+1, -1/x^k / (1 - 1/x^k + O(x^(n+2)) )^k + O(x^(n+2))); polcoeff(A, n)}

%o for(n=0, 80, print1(a(n), ", "))

%o (PARI) {a(n) = local(A=1); A = sum(k=1, sqrtint(n)+1, (-1)^(k-1) * x^(k^2-k)/(1-x^k)^k + O(x^(n+2))); polcoeff(A, n)}

%o for(n=0, 80, print1(a(n), ", "))

%Y Cf. A260116, A217668, A260147.

%K sign

%O 0,9

%A _Paul D. Hanna_, Jul 17 2015