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.

1, 2, -1, 4, -4, 1, 4, -10, 6, -1, 4, -20, 21, -8, 1, 4, -34, 56, -36, 10, -1, 4, -52, 125, -120, 55, -12, 1, 4, -74, 246, -329, 220, -78, 14, -1, 4, -100, 441, -784, 714, -364, 105, -16, 1, 4, -130, 736, -1680, 1992, -1364, 560, -136, 18, -1, 4, -164, 1161, -3312, 4950, -4356, 2379, -816, 171, -20, 1

Offset: 0

Author

Roger L. Bagula, May 23 2007

Keywords

• tabl
• sign

Comments

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

Examples

			Triangle begins:
  1;
  2,   -1;
  4,   -4,    1;
  4,  -10,    6,    -1;
  4,  -20,   21,    -8,    1;
  4,  -34,   56,   -36,   10,    -1;
  4,  -52,  125,  -120,   55,   -12,    1;
  4,  -74,  246,  -329,  220,   -78,   14,   -1;
  4, -100,  441,  -784,  714,  -364,  105,  -16,   1;
  4, -130,  736, -1680, 1992, -1364,  560, -136,  18,  -1;
  4, -164, 1161, -3312, 4950, -4356, 2379, -816, 171, -20, 1;
		

References

• R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 60.
• 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.

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A156608, A156609, A156610, A156612, A219233.

Programs

Maple

A129862 := proc(n,k)
    M := Matrix(n,n);
    for r from 1 to n do
    for c from 1 to n do
        if r = c then
            M[r,c] := 2;
        elif abs(r-c)= 1 then
            M[r,c] := -1;
        else
            M[r,c] := 0 ;
        end if;
    end do:
    end do:
    if n-2 >= 1 then
        M[n,n-2] := -1 ;
        M[n-2,n] := -1 ;
    end if;
    if n-1 >= 1 then
        M[n-1,n] := 0 ;
        M[n,n-1] := 0 ;
    end if;
    LinearAlgebra[CharacteristicPolynomial](M,x) ;
    (-1)^n*coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, May 31 2014

Mathematica

(* First program *)
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]]];
M[d_]:= Table[t[n,m,d], {n,1,d}, {m,1,d}];
p[n_, x_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
T[n_, k_]:= SeriesCoefficient[p[n, x], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 21 2021 *)
(* Second program *)
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 *)

Sage

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) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[1,2,-1]+flatten([T(n) for n in (2..12)]) # G. C. Greubel, Jun 21 2021

Formula

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