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A177727 a(0)=1; a(n) = a(n-1) * Fibonacci(3+n) * Fibonacci(1+n) / (Fibonacci(n))^2, n > 1.

%I A177727 #21 Feb 09 2025 02:46:07
%S A177727 1,3,30,180,1300,8736,60333,412335,2829310,19384200,132882696,
%T A177727 910735488,6242420665,42785803515,293259265950,2010026277756,
%U A177727 13776931957468,94428478367520,647222466507045,4436128656563175,30405678471399166,208403619747957648,1428419662108160400
%N A177727 a(0)=1; a(n) = a(n-1) * Fibonacci(3+n) * Fibonacci(1+n) / (Fibonacci(n))^2, n > 1.
%C A177727 Similar recurrences a(n) = a(n-1)*F(a0+n-1)*F(b0+n-1)/(F(n)*F(c0+n-1)) are:
%C A177727 {a0,b0,c0} = {3,2,1} in A066258.
%C A177727 {a0,b0,c0} = {3,1,1} in A001654.
%C A177727 {a0,b0,c0} = {4,1,1} in A001655 and next for 5,6 as well.
%D A177727 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, p. 93.
%H A177727 Vincenzo Librandi, <a href="/A177727/b177727.txt">Table of n, a(n) for n = 0..200</a>
%H A177727 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1)
%F A177727 G.f.: ( -1+2*x ) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - _R. J. Mathar_, Nov 17 2011
%F A177727 a(n) = A001656(n) - 2*A001656(n-1). - _R. J. Mathar_, Nov 17 2011
%p A177727 with (combinat):
%p A177727 A177727 := proc(n)
%p A177727    if n = 0 then
%p A177727            1;
%p A177727    else
%p A177727            procname(n-1)*fibonacci(3+n)*fibonacci(1+n)/fibonacci(n)^2 ;
%p A177727    end if;
%p A177727 end proc:
%p A177727 seq(A177727(n),n=0..10) ; # _R. J. Mathar_, Nov 17 2011
%t A177727 a0 = 4; b0 = 2; c0 = 1;
%t A177727 a[0] = 1;
%t A177727 a[n_] := a[n] = (Fibonacci[(a0 + n - 1)]*Fibonacci[( b0 + n - 1)]/(Fibonacci[n]*Fibonacci[(c0 + n - 1)]))*a[n - 1];
%t A177727 Table[a[n], {n, 0, 30}]
%t A177727 LinearRecurrence[{5,15,-15,-5,1},{1,3,30,180,1300},30] (* _Vincenzo Librandi_, Nov 18 2011 *)
%o A177727 (Magma) I:=[1, 3, 30, 180, 1300]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // _Vincenzo Librandi_, Nov 18 2011
%Y A177727 Cf. A066258, A001654, A001655, A001656, A001657.
%K A177727 nonn,easy
%O A177727 0,2
%A A177727 _Roger L. Bagula_, May 12 2010