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, 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

Author

Roger L. Bagula, Jul 31 2010

Keywords

• sign
• tabl
• easy

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.

Examples

			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;
		

Programs

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]

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).