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A109516 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1].

%I A109516 #33 Feb 28 2021 19:57:28
%S A109516 1,1,6,45,464,6000,93528,1707111,35721216,843160671,22165100000,
%T A109516 642268811184,20339749638144,698946255836933,25903663544572800,
%U A109516 1029945249481640625,43733528272753917952,1975222567881226040760
%N A109516 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 A109516 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 A109516 Seiichi Manyama, <a href="/A109516/b109516.txt">Table of n, a(n) for n = 1..387</a>
%F A109516 a(n+1) = [x^n] 1/(1 - n*x - n*x^2). - _Paul D. Hanna_, Dec 27 2012
%F A109516 a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*n^(n-k) for n>=0 (conjectured). - _Werner Schulte_, Oct 21 2016
%F A109516 a(n) = ((n + sqrt((n-1)*(n+3)) - 1)^n - (n - sqrt((n-1)*(n+3)) - 1)^n) / (2^n * sqrt((n-1)*(n+3))), for n > 1. - _Daniel Suteu_, Apr 20 2018
%F A109516 a(n) ~ n^(n-1). - _Vaclav Kotesovec_, Apr 20 2018
%F A109516 a(n+1) = (-sqrt(n)*i)^n * S(n, sqrt(n)*i) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. - _Seiichi Manyama_, Feb 28 2021
%e A109516 a(4)=45 because if M is the 2 X 2 matrix [0,1;3,3], then M^4 is the 2 X 2 matrix [36,45;135;171].
%e A109516 G.f. = x + x^2 + 6*x^3 + 45*x^4 + 464*x^5 + 6000*x^6 + 93528*x^7 + 1707111*x^8 + ...
%p A109516 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..20);
%t A109516 M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, n - 1}}, n], {{0, 1}, {1, 1}}]; a = Table[M[n][[1, 2]], {n, 1, 50}]
%t A109516 Table[SeriesCoefficient[1/(1 - n*x - n*x^2), {x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Apr 20 2018 *)
%o A109516 (PARI) {a(n)=polcoeff(1/(1-n*x-n*x^2+x*O(x^n)), n)} \\ _Paul D. Hanna_, Dec 27 2012
%o A109516 (PARI) a(n) = ([0,1;n-1,n-1]^n)[1, 2]; \\ _Michel Marcus_, Apr 20 2018
%o A109516 (PARI) a(n) = round((-sqrt(n-1)*I)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)*I/2)); \\ _Seiichi Manyama_, Feb 28 2021
%Y A109516 Cf. A000045, A000166, A109519.
%K A109516 nonn
%O A109516 1,3
%A A109516 _Roger L. Bagula_, Jun 16 2005