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A224415 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).

%I A224415 #7 Mar 27 2025 07:58:55
%S A224415 1,5,25,100,380,1348,4610,15250,49250,155860,485228,1489780,4520475,
%T A224415 13577775,40423155,119413496,350336200,1021523000,2962214500,
%U A224415 8547193700,24551057380,70231278200,200150437000,568435763000,1609247086325,4542394525369,12786764813645
%N A224415 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 A224415 Given g.f. A(x), note that A(x)^(1/5) does not yield an integer series.
%C A224415 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).
%H A224415 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-25,5,48,5,-25,0,5,-1).
%F A224415 G.f.: 1 / ( (1+x)^4 * (1 - 3*x + x^2)^3 ).
%e A224415 G.f.: 1 + 5*x + 25*x^2 + 100*x^3 + 380*x^4 + 1348*x^5 + 4610*x^6 +...
%e A224415 where
%e A224415 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 A224415 (PARI) {L(n)=fibonacci(n-1)^2+fibonacci(n+1)^2}
%o A224415 {a(n)=polcoeff(exp(sum(m=1, n, 5*L(m)*x^m/m)+x*O(x^n)), n)}
%o A224415 for(n=0, 30, print1((a(n)), ", "))
%o A224415 (PARI) {a(n)=polcoeff(1/((1+x)^4*(1-3*x+x^2)^3+x*O(x^n)),n)}
%o A224415 for(n=0, 30, print1((a(n)), ", "))
%Y A224415 Cf. A069921.
%K A224415 nonn
%O A224415 0,2
%A A224415 _Paul D. Hanna_, Apr 05 2013