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

1,2

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.

Links

Table of n, a(n) for n=1..29.

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).

Example

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)) +...

Prog

(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)}

Crossrefs

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

Sequence in context: A245488 A242604 A212150 * A097556 A196861 A211844

Adjacent sequences: A203835 A203836 A203837 * A203839 A203840 A203841

Keyword

nonn

Author

Paul D. Hanna, Jan 12 2012

Status

approved