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A237654 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n-1)*Fibonacci(n+1) * x^n/n ).

%I A237654 #20 Apr 17 2025 15:44:37
%S A237654 1,0,5,5,25,49,150,365,990,2550,6726,17550,46015,120390,315275,825299,
%T A237654 2160775,5656855,14809980,38772875,101508876,265753500,695751900,
%U A237654 1821501900,4768754125,12484760124,32685526625,85571819345,224029931845,586517975725,1535523995826,4020054011225,10524638038410
%N A237654 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n-1)*Fibonacci(n+1) * x^n/n ).
%C A237654 Compare to the g.f. of A054888.
%C A237654 Given g.f. A(x), note that A(x)^(1/5) is not an integer series.
%H A237654 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,5,0,-1).
%F A237654 G.f.: 1 / ( (1-3*x+x^2) * (1+x)^3 ).
%F A237654 a(n) = (2*Lucas(2*n+5) + (28+25*n+5*n^2)*(-1)^(n))/50 where Lucas = A000032. - _Greg Dresden_, Jan 01 2021
%e A237654 G.f.: A(x) = 1 + 5*x^2 + 5*x^3 + 25*x^4 + 49*x^5 + 150*x^6 + 365*x^7 + ...
%e A237654 where the logarithm begins:
%e A237654 log(A(x)) = 5*1*2*x^2/2 + 5*1*3*x^3/3 + 5*2*5*x^4/4 + 5*3*8*x^5/5 + 5*5*13*x^6/6 + 5*8*21*x^7/7 + 5*13*34*x^8/8 + ...
%t A237654 LinearRecurrence[{0,5,5,0,-1},{1,0,5,5,25},40] (* _Harvey P. Dale_, Apr 17 2025 *)
%o A237654 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, 5*fibonacci(m-1)*fibonacci(m+1)*x^m/m) + x*O(x^n)), n)}
%o A237654 for(n=0,36,print1(a(n),", "))
%Y A237654 Cf. A237655, A054888, A000032.
%K A237654 nonn
%O A237654 0,3
%A A237654 _Paul D. Hanna_, May 05 2014