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%I #5 Mar 30 2012 17:34:41
%S 1,5,-1,24,-10,1,115,-73,15,-1,551,-470,147,-20,1,2640,-2828,1190,
%T -246,25,-1,12649,-16310,8631,-2400,370,-30,1,60605,-91371,58275,
%U -20385,4225,-519,35,-1,290376,-501150,374115,-157800,41140,-6790,693,-40,1,1391275
%N Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.
%C The row sums are 1, 4, 15, 56, 209, 780, 2911, .. A001353.
%C Apart from signs, the same as A123967.
%C This can also be defined as the coefficients of the characteristic polynomial of the n X n tridiagonal symmetric matrix with 5's on the diagonal and -1's on the two adjacent subdiagonals. Expansion of the determinant along the first column yields the recurrence of the definition.
%F T(n,k) = 5*T(n-1,k)-T(n-1,k-1)-T(n-2,k) starting T(0,0)=1, T(1,0)=5 and T(1,1)=-1.
%F T(n,0) = A004254(n+1).
%e 1 ; # 1
%e 5, -1; # 5-x
%e 24, -10, 1 ; # 24-10x+x^2
%e 115, -73, 15, -1; # 115-73x+15x^2-x^3
%e 551, -470, 147, -20, 1;
%e 2640, -2828, 1190, -246, 25, -1;
%e 12649, -16310, 8631, -2400, 370, -30, 1;
%e 60605, -91371, 58275, -20385, 4225, -519, 35, -1;
%e 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1;
%e 1391275, -2704755, 2313450, -1142730, 359275, -74571, 10220, -892, 45, -1;
%t Clear[M, T, d, a, x, a0]
%t T[n_, m_, d_] := If[ n == m, 5, If[n == m - 1 || n == m + 1, -1, 0]]
%t M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]
%t Table[Det[M[d]], {d, 1, 10}]
%t Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]
%t a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], { d, 1, 10}]]
%t Flatten[a]
%t MatrixForm[a]
%K sign,tabl,easy
%O 0,2
%A _Roger L. Bagula_, Jul 31 2010
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