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A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

%I A203847 #24 Jul 18 2018 06:43:19
%S A203847 1,2,4,9,10,32,26,84,102,220,178,864,466,1508,2440,4935,3194,15504,
%T A203847 8362,40590,43784,70844,57314,370944,225075,485572,785672,1906866,
%U A203847 1028458,6656320,2692538,13069854,14098312,22811548,36909860,134373168,48315634,156352676,252983944
%N A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.
%C A203847 Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
%C A203847 Related identities:
%C A203847 (1) Sum_{n>=1} n^k*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Fibonacci(n)*x^n for k>=0.
%C A203847 (2) Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n.
%C A203847 (3) Sum_{n>=1} moebius(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = x.
%C A203847 (4) Sum_{n>=1} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^2)*x^(n^2).
%H A203847 G. C. Greubel, <a href="/A203847/b203847.txt">Table of n, a(n) for n = 1..2500</a>
%F A203847 G.f.: Sum_{n>=1} Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Fibonacci(n)*x^n, where Lucas(n) = A000204(n).
%e A203847 G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...
%e A203847 where A(x) = x/(1-x-x^2) + x^2/(1-3*x^2+x^4) + 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) + 5*x^5/(1-11*x^5-x^10) + 8*x^6/(1-18*x^6+x^12) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
%t A203847 Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* _G. C. Greubel_, Jul 17 2018 *)
%o A203847 (PARI) {a(n)=sigma(n,0)*fibonacci(n)}
%o A203847 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203847 {a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A203847 (PARI) a(n) = numdiv(n)*fibonacci(n); \\ _Michel Marcus_, Jul 18 2018
%Y A203847 Cf. A203848, A203849, A203838, A204060, A000005 (tau), A000204 (Lucas), A000045.
%Y A203847 Cf. A205507, A205882, A205963, A205964, A205965, A205966.
%Y A203847 Cf. A205967, A205968, A205969, A205970, A205971.
%Y A203847 Cf. A205972, A205973, A205974, A205975, A205976.
%Y A203847 Cf. A204291, A203801, A203850, A203860, A203861.
%K A203847 nonn
%O A203847 1,2
%A A203847 _Paul D. Hanna_, Jan 11 2012