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A260180 G.f.: Sum_{n>=0} x^n * (1 - x^n)^n.

%I A260180 #17 Mar 03 2025 13:31:44
%S A260180 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 A260180 -12,37,-48,1,45,1,-71,56,-16,36,-41,1,-18,79,-69,1,51,1,-186,232,-22,
%U A260180 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 A260180 G.f.: Sum_{n>=0} x^n * (1 - x^n)^n.
%C A260180 Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
%H A260180 Paul D. Hanna, <a href="/A260180/b260180.txt">Table of n, a(n) for n = 0..2050</a>
%F A260180 G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2-n) / (1 - x^n)^n.
%F A260180 G.f.: Sum_{n>=1} - x^(-n) / (1 - x^(-n))^n.
%F A260180 From  _Peter Bala_, Mar 02 2025: (Start)
%F A260180 For n >= 1, a(n) = Sum_{d divides n} (-1)^(d-1) * binomial(n/d, d-1).
%F A260180 For prime p > 3, a(p) = 1, a(2*p) = 1 - p and a(p^2) = p + 1. (End)
%e A260180 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 A260180 where
%e A260180 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 A260180 Also,
%e A260180 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 +...
%p A260180 with(numtheory):
%p A260180 seq(add((-1)^(d-1)*binomial(n/d, d-1), d in divisors(n)), n = 1..70); # _Peter Bala_, Mar 02 2025
%t A260180 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 A260180 (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 A260180 for(n=0, 80, print1(a(n), ", "))
%o A260180 (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 A260180 for(n=0, 80, print1(a(n), ", "))
%o A260180 (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 A260180 for(n=0, 80, print1(a(n), ", "))
%Y A260180 Cf. A260116, A217668, A260147.
%K A260180 sign,easy
%O A260180 0,9
%A A260180 _Paul D. Hanna_, Jul 17 2015