This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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