%I A158199
%S A158199 1,4,1,9,10,1,16,52,16,1,25,190,131,22,1,36,553,736,246,28,1,
%T A158199 49,1372,3206,1870,397,34,1,64,3024,11584,10900,3808,584,40,1,81,
%U A158199 6084,36234,52074,27847,6766,807,46,1,100,11385,101024,212738
%V A158199 1,4,-1,9,-10,1,16,-52,16,-1,25,-190,131,-22,1,36,-553,736,-246,28,-1,
%W A158199 49,-1372,3206,-1870,397,-34,1,64,-3024,11584,-10900,3808,-584,40,-1,81,
%X A158199 -6084,36234,-52074,27847,-6766,807,-46,1,100,-11385,101024,-212738
%N A158199 A coefficients of characteristic polynomials of A_n Cartan matrices times 
               their transposes: t(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m 
               + 1, -1, 0]]. M(d)=t(n,m,d)*Transpose[t(n,m,d)]
%C A158199 Row sums are:
%C A158199 {1, 3, 0, -21, -55, 0, 377, 987, 0, -6765, -17711,...},
%C A158199 Example matrix:
%C A158199 M(3)={{5, -4, 1},
%C A158199 {-4, 6, -4},
%C A158199 {1, -4, 5}}
%F A158199 t(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
%F A158199 M(d)=t(n,m,d)*Transpose[t(n,m,d)];
%F A158199 out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)
%e A158199 {1},
%e A158199 {4, -1},
%e A158199 {9, -10, 1},
%e A158199 {16, -52, 16, -1},
%e A158199 {25, -190, 131, -22, 1},
%e A158199 {36, -553, 736, -246, 28, -1},
%e A158199 {49, -1372, 3206, -1870, 397, -34, 1},
%e A158199 {64, -3024, 11584, -10900, 3808, -584, 40, -1},
%e A158199 {81, -6084, 36234, -52074, 27847, -6766, 807, -46, 1},
%e A158199 {100, -11385, 101024, -212738, 165844, -59555, 10960, -1066, 52, -1},
%e A158199 {121, -20086, 256399, -765314, 838915, -425272, 112828, -16606, 1361, 
               -58, 1}
%t A158199 Clear[M, T, d, a, x, a0];
%t A158199 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
%t A158199 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}].Transpose[Table[T[n, 
               m, d], {n, 1, d}, {m, 1, d}]];
%t A158199 a0 = Table[M[d], {d, 1, 10}];
%t A158199 Table[Det[M[d]], {d, 1, 10}];
%t A158199 Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
%t A158199 a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[ 
               n], x]], x], {n, 1, 10}]];
%t A158199 Flatten[a]
%t A158199 Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], 
               x]], x]], {n, 1, 10}]];
%Y A158199 Sequence in context: A021990 A084887 A067015 this_sequence A091885 A069606 
               A001254
%Y A158199 Adjacent sequences: A158196 A158197 A158198 this_sequence A158200 A158201 
               A158202
%K A158199 sign,tabl,uned
%O A158199 0,2
%A A158199 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 13 2009