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A109517 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,2(n-1)].

%I A109517 #10 Oct 19 2021 21:02:34
%S A109517 1,2,18,252,4880,120750,3639384,129365880,5298720768,245738908890,
%T A109517 12728860100000,728372947109940,45631105330876416,3106354479972026374,
%U A109517 228329428483544787840,18022862954171193750000,1520481402538463932186624,136531862779634547726146994
%N A109517 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,2(n-1)].
%C A109517 The (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;1,1] is the Fibonacci number A000045(n).
%H A109517 Robert Israel, <a href="/A109517/b109517.txt">Table of n, a(n) for n = 1..351</a>
%F A109517 For n > 1, a(n) = ((n - 1 + sqrt(n*(n - 1)))^n - (n - 1 - sqrt(n*(n - 1)))^n)/(2*sqrt(n*(n - 1))). - _Robert Israel_, Oct 19 2021
%e A109517 a(4)=252 because if M is the 2 X 2 matrix [0,1;3,6], then M^4 is the 2 X 2 matrix [117,252;756;1629].
%p A109517 with(linalg): a:=proc(n) local A,k: A[1]:=matrix(2,2,[0,1,n-1,2*(n-1)]): for k from 2 to n do A[k]:=multiply(A[k-1],A[1]) od: A[n][1,2] end: seq(a(n),n=1..19);
%p A109517 # second Maple program:
%p A109517 a:= n-> (<<0|1>, <n-1|2*n-2>>^n)[1, 2]:
%p A109517 seq(a(n), n=1..18);  # _Alois P. Heinz_, Oct 19 2021
%t A109517 M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, 2*(n - 1)}}, n], {{0, 1}, {1, 1}}] a = Table[M[n][[1, 2]], {n, 1, 50}]
%Y A109517 Cf. A000045, A000166.
%K A109517 nonn
%O A109517 1,2
%A A109517 _Roger L. Bagula_, Jun 16 2005