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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
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.
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
KEYWORD
sign,tabl,easy
AUTHOR
Roger L. Bagula, Jul 31 2010
STATUS
approved