A261608 - OEIS

A261608

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

2, 1, 4, 5, 6, 6, 8, 16, 12, 15, 12, 32, 14, 28, 32, 52, 18, 55, 20, 74, 72, 66, 24, 160, 28, 91, 140, 146, 30, 205, 32, 271, 244, 153, 72, 442, 38, 190, 392, 563, 42, 518, 44, 505, 788, 276, 48, 1510, 52, 451, 852, 896, 54, 1086, 728, 1748, 1180, 435, 60, 3291, 62, 496, 1648, 2867, 1848, 2101, 68, 2481, 2072, 1953, 72, 7634

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

FORMULA

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

G.f.: Sum_{n=-oo..+oo} x^(n^2)/(1 + x^n)^n, where the sum is taken to exclude the coefficient of x^0.

G.f.: Sum_{n=-oo..+oo} (1 + x^n)^n, where the sum is taken to exclude the coefficient of x^0.

G.f.: x * d/dx Sum_{n=-oo..+oo, n<>0} (1/n^2) * x^(n^2)/(1 - x^n)^n. - Paul D. Hanna, Nov 16 2017

EXAMPLE

G.f.: A(x) = 2*x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 8*x^7 + 16*x^8 + 12*x^9 + 15*x^10 + 12*x^11 + 32*x^12 + 14*x^13 + 28*x^14 +...

where A(x) = N(x) + P(x) such that

N(x) = x*(x-1)^0 + x^2*(x^2-1) + x^3*(x^3-1)^2 + x^4*(x^4-1)^3 + x^5*(x^5-1)^4 + x^6*(x^6-1)^5 + x^7*(x^7-1)^6 + x^8*(x^8-1)^7 +...

P(x) = x/(1-x)^2 + x^4/(1-x^2)^3 + x^9/(1-x^3)^4 + x^16/(1-x^4)^5 + x^25/(1-x^5)^6 + x^36/(1-x^6)^7 + x^49/(1-x^7)^8 +...

explicitly,

N(x) = x - x^2 + x^3 + x^5 - 3*x^6 + x^7 + 2*x^8 + 2*x^9 - 5*x^10 + x^11 + x^12 + x^13 - 7*x^14 + 7*x^15 + 7*x^16 + x^17 - 19*x^18 + x^19 + 4*x^20 +...

P(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 +...

PROG

(PARI) {a(n) = polcoeff(sum(m=-n-1, n+1, if(m!=0, x^(m^2)/(1-x^m +x*O(x^n))^(m+1))), n)}

for(n=1, 60, print1(a(n), ", "))

(PARI) {a(n) = polcoeff(sum(m=-n-1, n+1, if(m!=0, x^m*(x^m-1 +x*O(x^n))^(m-1))), n)}

for(n=1, 60, print1(a(n), ", "))

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

for(n=1, 60, print1(a(n), ", "))

(PARI) {a(n) = polcoeff(sum(m=-n-1, n+1, (1 + x^m +x*O(x^n))^m), n)}

for(n=1, 60, print1(a(n), ", "))

CROSSREFS

Cf. A261605.