A326607 G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.

1, 3, 2, 7, -1, 11, -3, 15, -19, 40, -34, 23, -6, 27, -76, 141, -139, 35, 61, 39, -270, 358, -208, 47, -35, 106, -298, 739, -874, 59, 725, 63, -1415, 1332, -526, 596, -617, 75, -664, 2185, -2069, 83, 1519, 87, -4696, 5740, -988, 95, -3571, 204, 3236, 4863, -8995, 107, 1891, 7701, -11578, 6784, -1594, 119, -941

Offset

0,2

Comments

Compare the g.f. to the series: Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)) = Product_{n>=1} (1 - x^(2*n))^3.

Compare the g.f. to the series: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2550

Formula

G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.

G.f.: Sum_{n>=0} (-1)^n * (2*n+1 + x^(n+1)) * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

Example

G.f.: A(x) = 1 + 3*x + 2*x^2 + 7*x^3 - x^4 + 11*x^5 - 3*x^6 + 15*x^7 - 19*x^8 + 40*x^9 - 34*x^10 + 23*x^11 - 6*x^12 + 27*x^13 - 76*x^14 + 141*x^15 + ...
where A(x) is equal to the series given by
A(x) = 1 + 3*x*(1-x) + 5*x^2*(1-x^2)^2 + 7*x^3*(1-x^3)^3 + 9*x^4*(1-x^4)^4 + 11*x^5*(1-x^5)^5 + 13*x^6*(1-x^6)^6 + 15*x^7*(1-x^7)^7 + ...
Also,
A(x) = (1 + x)/(1 - x)^2 - (3 + x^2)*x^2/(1 - x^2)^3 + (5 + x^3)*x^6/(1 - x^3)^4 - (7 + x^4)*x^12/(1 - x^4)^5 + (9 + x^5)*x^20/(1 - x^5)^6 - (11 + x^6)*x^30/(1 - x^6)^7 + (13 + x^7)*x^42/(1 - x^7)^8 + ...

Prog

(PARI) /* By definition */
{a(n) = my(A = sum(m=0, n, (2*m + 1) * x^m * (1 - x^m + x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) /* Accelerated series */
{a(n) = my(A = sum(m=0, sqrtint(n+1), (-1)^m * (2*m + 1 + x^(m+1))* x^(m*(m+1)) / (1 - x^(m+1) + x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A326605, A260180.

Sequence in context: A010760 A085594 A085587 * A071189 A137822 A300845

Adjacent sequences: A326604 A326605 A326606 * A326608 A326609 A326610

Keyword

sign

Author

Paul D. Hanna, Oct 08 2019

Status

approved