0,1

2*T(2*n,x) = sum(a(n,m)*(2*x)^(2*m),m=0..n).

Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).

Comment from njas, Jun 09 2007: if the leading 2 is replaced by a 1 we get the essentially identical sequence A110162.

Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 23 2007

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62

Sigurdur Helgasson,Differential Geometry, Lie Groups, and Symmetric Spaces,Graduaste Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463

W. Lang, First 10 rows and more.

Eric Weisstein's World of Mathematics, Cartan Matrix

Eric Weisstein's World of Mathematics, Dynkin Diagram

a(n,m)=0 if n<m; a(n,0)=2*(-1)^n; a(n,m)=((-1)^(n+m))*n*binomial(n+m-1,2*m-1)/m.

[2]; [ -2,1]; [2,-4,1]; [ -2,9,-6,1]; [2,-16,20,-8,1]; ...

n=3: [ -2,9,-6,1] stands for -2*1 + 9*(2*x)^2 -6*(2*x)^4 +1*(2*x)^6 =2*(1+18*x^2-48*x^4+32*x^6) = 2*T(6,x).

T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -2, If[(n == m - 1 || n == m + 1), -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a = Join[M[1], Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10} ]] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 23 2007

Row sums (signed): -A061347(n+3), n>=0. Row sums (unsigned): A005248(n)=L(2*n) (Lucas).

Cf. A005248, A053122.

Sequence in context: A022479 A021456 A125912 this_sequence A007427 A048106 A056671

Adjacent sequences: A127674 A127675 A127676 this_sequence A127678 A127679 A127680

sign,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007