A207834 - OEIS

%I #14 Mar 30 2012 18:37:37

%S 1,5,25,130,1295,38861,4227075,1309117220,1123176929475,

%T 2564594183278115,15604715134340991949,251021373648740285348860,

%U 10668788238489683954523431475,1195322752666989652479885363067075,352750492054485236937115646128341734205

%N G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

%C Given g.f. A(x), note that A(x)^(1/5) is not an integer series.

%C Compare the definition to the g.f. of the Fibonacci numbers:

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

%e G.f.: A(x) = 1 + 5*x + 25*x^2 + 130*x^3 + 1295*x^4 + 38861*x^5 +...

%e such that, by definition,

%e log(A(x))/5 = x + 5*x^2/2 + 28*x^3/3 + 641*x^4/4 + 33011*x^5/5 +...+ (Fibonacci(n-1)^n + Fibonacci(n+1)^n)*x^n/n +...

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

%o {a(n)=polcoeff(exp(sum(m=1,n,5*L(m)*x^m/m)+x*O(x^n)),n)}

%o for(n=0,51,print1(a(n),", "))

%Y Cf. A207835, A156216, A166168.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 20 2012