%I #13 Jul 18 2018 03:02:14

%S 1,5,20,63,130,400,650,1785,3094,7150,10858,30240,39610,94250,158600,

%T 336567,463130,1175720,1513522,3693690,5473000,10803710,15188210,

%U 39412800,48841275,103184050,161062760,333701550,432980818,1081652000,1295110778,2973391785,4299985160

%N a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

%C Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

%H G. C. Greubel, <a href="/A203849/b203849.txt">Table of n, a(n) for n = 1..2500</a>

%F G.f.: Sum_{n>=1} n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).

%e G.f.: A(x) = x + 5*x^2 + 20*x^3 + 63*x^4 + 130*x^5 + 400*x^6 + 650*x^7 +...

%e where A(x) = x/(1-x-x^2) + 2^2*1*x^2/(1-3*x^2+x^4) + 3^2*2*x^3/(1-4*x^3-x^6) + 4^2*3*x^4/(1-7*x^4+x^8) + 5^2*5*x^5/(1-11*x^5-x^10) + 6^2*8*x^6/(1-18*x^6+x^12) +...+ n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

%t Table[DivisorSigma[2, n]*Fibonacci[n], {n, 50}] (* _G. C. Greubel_, Jul 17 2018 *)

%o (PARI) {a(n)=sigma(n,2)*fibonacci(n)}

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

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

%Y Cf. A203847, A203848, A203838, A001157 (sigma_2), A000204 (Lucas), A000045.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 12 2012