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

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

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.

From Peter Bala, Mar 02 2025: (Start)

For n >= 1, a(n) = Sum_{d divides n} (-1)^(d-1) * binomial(n/d, d-1).

For prime p > 3, a(p) = 1, a(2*p) = 1 - p and a(p^2) = p + 1. (End)

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 +...

Maple

with(numtheory):
seq(add((-1)^(d-1)*binomial(n/d, d-1), d in divisors(n)), n = 1..70); # Peter Bala, Mar 02 2025

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

Cf. A260116, A217668, A260147.

Sequence in context: A177935 A021748 A132723 * A057279 A350908 A054733

Adjacent sequences: A260177 A260178 A260179 * A260181 A260182 A260183

Keyword

sign,easy,changed

Author

Paul D. Hanna, Jul 17 2015

Status

approved