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A129862 Triangle read by rows: T(n,k) is the coefficient [x^k] of (-1)^n times the characteristic polynomial of the Cartan matrix for the root system D_n.

%I A129862 #30 Sep 20 2024 23:54:08
%S A129862 1,2,-1,4,-4,1,4,-10,6,-1,4,-20,21,-8,1,4,-34,56,-36,10,-1,4,-52,125,
%T A129862 -120,55,-12,1,4,-74,246,-329,220,-78,14,-1,4,-100,441,-784,714,-364,
%U A129862 105,-16,1,4,-130,736,-1680,1992,-1364,560,-136,18,-1,4,-164,1161,-3312,4950,-4356,2379,-816,171,-20,1
%N A129862 Triangle read by rows: T(n,k) is the coefficient [x^k] of (-1)^n times the characteristic polynomial of the Cartan matrix for the root system D_n.
%C A129862 Row sums of the absolute values are s(n) = 1, 3, 9, 21, 54, 141, 369, 966, 2529, 6621, 17334, ... (i.e., s(n) = 3*|A219233(n-1)| for n > 0). - _R. J. Mathar_, May 31 2014
%D A129862 R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 60.
%D A129862 Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978, p. 464.
%H A129862 G. C. Greubel, <a href="/A129862/b129862.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A129862 T(n, k) = coefficients of ( (2-x)*Lucas(2*n-2, i*sqrt(x)) ) with T(0, 0) = 1, T(1, 0) = 2 and T(1, 1) = -1. - _G. C. Greubel_, Jun 21 2021
%e A129862 Triangle begins:
%e A129862   1;
%e A129862   2,   -1;
%e A129862   4,   -4,    1;
%e A129862   4,  -10,    6,    -1;
%e A129862   4,  -20,   21,    -8,    1;
%e A129862   4,  -34,   56,   -36,   10,    -1;
%e A129862   4,  -52,  125,  -120,   55,   -12,    1;
%e A129862   4,  -74,  246,  -329,  220,   -78,   14,   -1;
%e A129862   4, -100,  441,  -784,  714,  -364,  105,  -16,   1;
%e A129862   4, -130,  736, -1680, 1992, -1364,  560, -136,  18,  -1;
%e A129862   4, -164, 1161, -3312, 4950, -4356, 2379, -816, 171, -20, 1;
%p A129862 A129862 := proc(n,k)
%p A129862     M := Matrix(n,n);
%p A129862     for r from 1 to n do
%p A129862     for c from 1 to n do
%p A129862         if r = c then
%p A129862             M[r,c] := 2;
%p A129862         elif abs(r-c)= 1 then
%p A129862             M[r,c] := -1;
%p A129862         else
%p A129862             M[r,c] := 0 ;
%p A129862         end if;
%p A129862     end do:
%p A129862     end do:
%p A129862     if n-2 >= 1 then
%p A129862         M[n,n-2] := -1 ;
%p A129862         M[n-2,n] := -1 ;
%p A129862     end if;
%p A129862     if n-1 >= 1 then
%p A129862         M[n-1,n] := 0 ;
%p A129862         M[n,n-1] := 0 ;
%p A129862     end if;
%p A129862     LinearAlgebra[CharacteristicPolynomial](M,x) ;
%p A129862     (-1)^n*coeftayl(%,x=0,k) ;
%p A129862 end proc: # _R. J. Mathar_, May 31 2014
%t A129862 (* First program *)
%t A129862 t[n_, m_, d_]:= If[n==m, 2, If[(m==d && n==d-2) || (n==d && m==d-2), -1, If[(n==m- 1 || n==m+1) && n<=d-1 && m<=d-1, -1, 0]]];
%t A129862 M[d_]:= Table[t[n,m,d], {n,1,d}, {m,1,d}];
%t A129862 p[n_, x_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
%t A129862 T[n_, k_]:= SeriesCoefficient[p[n, x], {x, 0, k}];
%t A129862 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 21 2021 *)
%t A129862 (* Second program *)
%t A129862 Join[{{1}, {2, -1}}, CoefficientList[Table[(2-x)*LucasL[2(n-1), Sqrt[-x]], {n, 2, 10}], x]]//Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)
%o A129862 (Sage)
%o A129862 def p(n,x): return 2*(2-x)*sum( ((n-1)/(2*n-k-2))*binomial(2*n-k-2, k)*(-x)^(n-k-1) for k in (0..n-1) )
%o A129862 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A129862 [1,2,-1]+flatten([T(n) for n in (2..12)]) # _G. C. Greubel_, Jun 21 2021
%Y A129862 Cf. A156608, A156609, A156610, A156612, A219233.
%K A129862 tabl,sign
%O A129862 0,2
%A A129862 _Roger L. Bagula_, May 23 2007