A203838 a(n) = sigma_3(n)*Fibonacci(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

1, 9, 56, 219, 630, 2016, 4472, 12285, 25738, 62370, 118548, 294336, 512134, 1167192, 2152080, 4620147, 7847658, 17604792, 28681660, 62224470, 105431872, 212319468, 348698376, 759507840, 1181718775, 2401396326, 4014783920, 7980869832, 12542045310

Offset: 1

Author

Paul D. Hanna, Jan 12 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^3*x^n/(1-x^n) = Sum_{n>=1} sigma_3(n)*x^n.

Examples

			G.f.: A(x) = x + 9*x^2 + 56*x^3 + 219*x^4 + 630*x^5 + 2016*x^6 +...
where A(x) = x/(1-x-x^2) + 2^3*1*x^2/(1-3*x^2+x^4) + 3^3*2*x^3/(1-4*x^3-x^6) + 4^3*3*x^4/(1-7*x^4+x^8) + 5^3*5*x^5/(1-11*x^5-x^10) + 6^3*8*x^6/(1-18*x^6+x^12) +...+ n^3*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
		

Crossrefs

Cf. A203847, A203848, A203849, A001158 (sigma_3), A000204 (Lucas), A000045.

Programs

PARI

{a(n)=sigma(n,3)*fibonacci(n)}

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,m^3*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

Formula

G.f.: Sum_{n>=1} n^3*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).