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A203848 a(n) = sigma(n)*Fibonacci(n), where sigma(n) = A000203(n), the sum of divisors of n.

%I A203848 #16 Sep 08 2022 08:46:01
%S A203848 1,3,8,21,30,96,104,315,442,990,1068,4032,3262,9048,14640,30597,28746,
%T A203848 100776,83620,284130,350272,637596,687768,2782080,2325775,5098506,
%U A203848 7856720,17797416,15426870,59906880,43080608,137233467,169179744,307955898,442918320,1358662032
%N A203848 a(n) = sigma(n)*Fibonacci(n), where sigma(n) = A000203(n), the sum of divisors of n.
%C A203848 Compare g.f. to the Lambert series identity: Sum_{n>=1} n*x^n/(1-x^n) = Sum_{n>=1} sigma(n)*x^n.
%H A203848 Vincenzo Librandi, <a href="/A203848/b203848.txt">Table of n, a(n) for n = 1..1000</a>
%F A203848 G.f.: Sum_{n>=1} n*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).
%e A203848 G.f.: A(x) = x + 3*x^2 + 8*x^3 + 21*x^4 + 30*x^5 + 96*x^6 + 104*x^7 +...
%e A203848 where A(x) = x/(1-x-x^2) + 2*1*x^2/(1-3*x^2+x^4) + 3*2*x^3/(1-4*x^3-x^6) + 4*3*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 6*8*x^6/(1-18*x^6+x^12) +...+ n*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
%t A203848 Table[DivisorSigma[1, n] Fibonacci[n], {n, 40}] (* _Wesley Ivan Hurt_, Aug 10 2016 *)
%o A203848 (PARI) {a(n)=sigma(n)*fibonacci(n)}
%o A203848 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203848 {a(n)=polcoeff(sum(m=1,n,m*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A203848 (Magma) [DivisorSigma(1, n)*Fibonacci(n): n in [1..40]]; // _Vincenzo Librandi_, Aug 12 2016
%Y A203848 Cf. A203847, A203849, A203838, A000203 (sigma), A000204 (Lucas), A000045.
%K A203848 nonn,easy
%O A203848 1,2
%A A203848 _Paul D. Hanna_, Jan 12 2012