A203847 - OEIS

%I #24 Jul 18 2018 06:43:19

%S 1,2,4,9,10,32,26,84,102,220,178,864,466,1508,2440,4935,3194,15504,

%T 8362,40590,43784,70844,57314,370944,225075,485572,785672,1906866,

%U 1028458,6656320,2692538,13069854,14098312,22811548,36909860,134373168,48315634,156352676,252983944

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

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

%C Related identities:

%C (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 (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 (3) Sum_{n>=1} moebius(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = x.

%C (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 G. C. Greubel, <a href="/A203847/b203847.txt">Table of n, a(n) for n = 1..2500</a>

%F 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 G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...

%e 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 Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* _G. C. Greubel_, Jul 17 2018 *)

%o (PARI) {a(n)=sigma(n,0)*fibonacci(n)}

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

%o {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 (PARI) a(n) = numdiv(n)*fibonacci(n); \\ _Michel Marcus_, Jul 18 2018

%Y Cf. A203848, A203849, A203838, A204060, A000005 (tau), A000204 (Lucas), A000045.

%Y Cf. A205507, A205882, A205963, A205964, A205965, A205966.

%Y Cf. A205967, A205968, A205969, A205970, A205971.

%Y Cf. A205972, A205973, A205974, A205975, A205976.

%Y Cf. A204291, A203801, A203850, A203860, A203861.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 11 2012