A260148 Expansion of Sum_{n>=0} x^(n^2-n) / (1 + x^n)^n.

1, -1, 2, -1, -1, -1, 5, -1, -3, -4, 6, -1, 2, -1, 8, -11, -11, -1, 25, -1, 2, -22, 12, -1, -3, -6, 14, -37, 22, -1, 77, -1, -71, -56, 18, -36, 127, -1, 20, -79, -69, -1, 135, -1, 144, -232, 24, -1, -179, -8, 236, -137, 261, -1, 307, -331, -362, -172, 30, -1, 859, -1, 32, -295, -599, -716, 727, -1, 647, -254, 1247

Offset: 0

Author

Paul D. Hanna, Jul 17 2015

Keywords

• sign

Examples

			A(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 +...
where
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 + ...
Also,
A(x) = 1 + x*(x-1) + x^2*(x^2-1)^2 + x^3*(x^3-1)^3 + x^4*(x^4-1)^4 + x^5*(x^5-1)^5 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2050

Crossrefs

Cf. A217668, A260147.

Programs

PARI

a(n) = local(A=1); A = sum(k=1, sqrtint(n)+1, x^(k^2-k) / (1 + x^k +x*O(x^n) )^k ); polcoeff(A, n);
for(n=0, 70, print1(a(n), ", "))

PARI

a(n) = local(A=0); A = sum(k=1,n+1, x^(-k)/(1 + x^(-k) +x*O(x^n) )^k ); polcoeff(A, n);
for(n=0, 70, print1(a(n), ", "))

PARI

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-n/d+1)*binomial(d, n/d-1))); \\ Seiichi Manyama, Feb 20 2023

Formula

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

G.f.: Sum_{n>=0} x^n * (x^n - 1)^n. - Paul D. Hanna, May 30 2018

a(p) = -1 for odd primes p.

a(n) = Sum_{d|n} (-1)^(d-n/d+1) * binomial(d,n/d-1) for n > 0. - Seiichi Manyama, Feb 20 2023