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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

Table of n, a(n) for n=0..45.

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 October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)