A109519 - OEIS

%I #23 Mar 01 2021 02:00:47

%S -1,-1,-2,-9,-80,-1000,-15336,-276115,-5705728,-133155495,-3464900000,

%T -99490865760,-3125217447936,-106614813012877,-3925516139359360,

%U -155164259295703125,-6553564019985219584,-294562012662334323872,-14038370700094085018112

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

%C 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).

%F From _Seiichi Manyama_, Feb 28 2021: (Start)

%F a(n+1) = [x^n] 1/(-1 + n*x - n*x^2).

%F a(n+1) = (-1)^(n+1) * Sum_{k=0..n} (-n)^k * binomial(k,n-k).

%F a(n+1) = (-1) * sqrt(n)^n * S(n, sqrt(n)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. (End)

%e a(4)=-9 because if M is the 2 X 2 matrix [0,-1;3,3], then M^4 is the 2 X 2 matrix [ -18,-9,27,9].

%p with(linalg): a:=proc(n) local A,k: A[1]:=matrix(2,2,[0,-1,n-1,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..21);

%t M[n_] = If[n > 1, MatrixPower[{{0, -1}, {n - 1, (n - 1)}}, n], {{0, 1}, {1, 1}}] a = Table[Abs[M[n][[1, 2]]], {n, 1, 50}]

%o (Sage) [ -lucas_number1(n+1,n,n) for n in range(0,19)] # _Zerinvary Lajos_, Jul 16 2008

%o (PARI) a(n) = round(-sqrt(n-1)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)/2)); \\ _Seiichi Manyama_, Feb 28 2021

%Y Cf. A000045, A000166, A109516, A109517, A109518.

%K sign

%O 1,3

%A _Roger L. Bagula_, Jun 16 2005