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A136678 Triangle read by rows: n-th row (n>=0) gives coefficients of characteristic polynomial of n X n generalized Cartan matrix M defined in Comments. 7
1, 2, -1, 2, -4, 1, 2, -9, 6, -1, 1, -16, 20, -8, 1, 0, -24, 50, -35, 10, -1, -2, -32, 104, -112, 54, -12, 1, -4, -38, 190, -293, 210, -77, 14, -1, -7, -40, 314, -664, 659, -352, 104, -16, 1, -10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1, -14, -20, 677, -2512, 4234, -3992, 2274, -800, 170, -20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

M has 2's on the main diagonal, -1's on the two adjacent diagonals, except that the entry M[floor(n/2)+1, floor(n/2)] is -2.

For n=4, M is the Cartan matrix of type F_4.

Row sums are 1, 1, -1, -2, -2, 0, 1, 1, -1, -2, -2, ...

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n, m, d)= If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]

EXAMPLE

Triangle begins:

{1},

{2, -1},

{2, -4, 1},

{2, -9, 6, -1},

{1, -16, 20, -8,1},

{0, -24, 50, -35, 10, -1},

{-2, -32, 104, -112, 54, -12, 1},

{-4, -38, 190, -293, 210, -77, 14, -1},

{-7, -40, 314, -664, 659, -352, 104, -16, 1},

{-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},

{-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}

...

For n=4, the matrix M is

[2,-1,0,0],

[-1,2,-1,0],

[0,-2,2,-1],

[0,0,-1,2],

which has charpoly x^4-8*x^3+20*x^2-16*x+1. The coefficients (starting with the constant term) give row 4 of the triangle.

MATHEMATICA

T[n_, m_, d_] := If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a0 = Table[M[d], {d, 1, 10}]; Table[Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] MatrixForm[a];

CROSSREFS

Sequence in context: A080246 A113413 A125694 * A110162 A199087 A306913

Adjacent sequences:  A136675 A136676 A136677 * A136679 A136680 A136681

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Apr 05 2008

EXTENSIONS

Edited by N. J. A. Sloane, Jan 27 2014

STATUS

approved

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Last modified October 15 07:56 EDT 2019. Contains 328026 sequences. (Running on oeis4.)