OFFSET
0,9
COMMENTS
Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2050
FORMULA
G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2-n) / (1 - x^n)^n.
G.f.: Sum_{n>=1} - x^(-n) / (1 - x^(-n))^n.
EXAMPLE
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 +...
where
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 +...
Also,
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 +...
MATHEMATICA
terms = 100; 1 + Sum[x^n*(1 - x^n)^n, {n, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
PROG
(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)}
for(n=0, 80, print1(a(n), ", "))
(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)}
for(n=0, 80, print1(a(n), ", "))
(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)}
for(n=0, 80, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 17 2015
STATUS
approved