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