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 A179900 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. 1
 1, 5, -1, 24, -10, 1, 115, -73, 15, -1, 551, -470, 147, -20, 1, 2640, -2828, 1190, -246, 25, -1, 12649, -16310, 8631, -2400, 370, -30, 1, 60605, -91371, 58275, -20385, 4225, -519, 35, -1, 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1, 1391275 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The row sums are 1, 4, 15, 56, 209, 780, 2911, .. A001353. Apart from signs, the same as A123967. 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. LINKS FORMULA 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. T(n,0) = A004254(n+1). EXAMPLE 1 ;       # 1 5, -1;     # 5-x 24, -10, 1 ;  # 24-10x+x^2 115, -73, 15, -1; # 115-73x+15x^2-x^3 551, -470, 147, -20, 1; 2640, -2828, 1190, -246, 25, -1; 12649, -16310, 8631, -2400, 370, -30, 1; 60605, -91371, 58275, -20385, 4225, -519, 35, -1; 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1; 1391275, -2704755, 2313450, -1142730, 359275, -74571, 10220, -892, 45, -1; MATHEMATICA Clear[M, T, d, a, x, a0] T[n_, m_, d_] := If[ n == m, 5, If[n == m - 1 || n == m + 1, -1, 0]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], { d, 1, 10}]] Flatten[a] MatrixForm[a] CROSSREFS Sequence in context: A201884 A294138 A207824 * A123967 A162259 A077195 Adjacent sequences:  A179897 A179898 A179899 * A179901 A179902 A179903 KEYWORD sign,tabl,easy AUTHOR Roger L. Bagula, Jul 31 2010 STATUS approved

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Last modified September 28 04:53 EDT 2021. Contains 347703 sequences. (Running on oeis4.)