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%I #5 Apr 06 2013 11:40:22
%S 1,5,25,100,380,1348,4610,15250,49250,155860,485228,1489780,4520475,
%T 13577775,40423155,119413496,350336200,1021523000,2962214500,
%U 8547193700,24551057380,70231278200,200150437000,568435763000,1609247086325,4542394525369,12786764813645
%N G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ) where L(n) = Fibonacci(n-1)^2 + Fibonacci(n+1)^2 = A069921(n-1).
%C Given g.f. A(x), note that A(x)^(1/5) does not yield an integer series.
%C Compare to: exp( Sum_{n>=1} Lucas(n)*x^n/n ) = 1/(1-x-x^2) where Lucas(n) = Fibonacci(n-1) + Fibonacci(n+1).
%F G.f.: 1 / ( (1+x)^4 * (1 - 3*x + x^2)^3 ).
%e G.f.: 1 + 5*x + 25*x^2 + 100*x^3 + 380*x^4 + 1348*x^5 + 4610*x^6 +...
%e where
%e log(A(x))/5 = x + 5*x^2/2 + 10*x^3/3 + 29*x^4/4 + 73*x^5/5 + 194*x^6/6 + 505*x^7/7 + 1325*x^8/8 +...+ A069921(n-1)*x^n/n +...
%o (PARI) {L(n)=fibonacci(n-1)^2+fibonacci(n+1)^2}
%o {a(n)=polcoeff(exp(sum(m=1, n, 5*L(m)*x^m/m)+x*O(x^n)), n)}
%o for(n=0, 30, print1((a(n)), ", "))
%o (PARI) {a(n)=polcoeff(1/((1+x)^4*(1-3*x+x^2)^3+x*O(x^n)),n)}
%o for(n=0, 30, print1((a(n)), ", "))
%Y Cf. A069921.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 05 2013
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