A261605 G.f.: Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n)^n.

1, 2, -1, 4, -1, 6, -6, 8, 2, 12, -15, 12, 8, 14, -28, 32, 22, 18, -55, 20, 34, 72, -66, 24, 44, 28, -91, 140, 62, 30, -205, 32, 209, 244, -153, 72, -98, 38, -190, 392, 443, 42, -518, 44, -1, 788, -276, 48, 506, 52, -451, 852, -196, 54, -1086, 728, 1636, 1180, -435, 60, -1691

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

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

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

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

Example

G.f.: A(x) = 1 + 2*x - x^2 + 4*x^3 - x^4 + 6*x^5 - 6*x^6 + 8*x^7 + 2*x^8 + 12*x^9 - 15*x^10 + 12*x^11 + 8*x^12 + 14*x^13 - 28*x^14 + 32*x^15 + 22*x^16 +...
where A(x) = 1 + N(x) + P(x) such that
N(x) = (x-1) + (x^2-1)^2 + (x^3-1)^3 + (x^4-1)^4 + (x^5-1)^5 + (x^6-1)^6 +...
P(x) = x/(1-x) + x^4/(1-x^2)^2 + x^9/(1-x^3)^3 + x^16/(1-x^4)^4 + x^25/(1-x^5)^5 +...
explicitly,
N(x) = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 2*x^8 + 10*x^9 - 20*x^10 + 11*x^11 - x^12 + 13*x^13 - 35*x^14 + 25*x^15 + 13*x^16 +...
P(x) = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 5*x^10 + x^11 + 9*x^12 + x^13 + 7*x^14 + 7*x^15 + 9*x^16 +...+ A143862(n)*x^n +...

Prog

(PARI) {a(n) = polcoeff(sum(m=-n-2, n+2, x^(m^2)/(1-x^m +x*O(x^n))^m), n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = polcoeff(sum(m=-n-2, n+2, (x^m-1 +x*O(x^n))^m), n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = polcoeff(1/2 + sum(m=-n-2, n+2, x^(m^2)/(1+x^m +x*O(x^n))^(m+1)), n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n) = polcoeff(1/2 + sum(m=-n-2, n+2, x^m*(1+x^m +x*O(x^n))^(m-1)), n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A143862, A260147, A261608.

Sequence in context: A305098 A128099 A182242 * A239093 A263453 A108952

Adjacent sequences: A261602 A261603 A261604 * A261606 A261607 A261608

Keyword

sign

Author

Paul D. Hanna, Aug 25 2015

Status

approved