A126569 Top-left "head" entry of the n-th power of the E8 Cartan matrix.

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867146, 3269060, 12446684, 47771496, 184544427, 716658870, 2794956099, 10938266562, 42930256917, 168890693650, 665739119129, 2628578437646, 10393091551794, 41141896235012, 163028816478833

Offset: 0

Author

Gary W. Adamson, Dec 28 2006

Keywords

• nonn

Examples

			a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0,0].
		

Links

• Wikipedia, E8
• Index entries for linear recurrences with constant coefficients, signature (16,-105,364,-714,784,-440,96,-1).

Crossrefs

Cf. A126566, A126567, A126568.

Programs

Maple

E8 := matrix(8,8,[ [2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, -1 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 2 ] ]) ;
printf("1,") ; for n from 1 to 20 do T := evalm(E8^n) ; printf("%a,", T[1,1]) ; od: # R. J. Mathar, May 08 2009

Formula

a(n) = leftmost term in M^n * [1,0,0,0,0,0,0,0], where M = the 8x8 matrix [2,-1,0,0,0,0,0,0; -1,2,-1,0,0,0,0,0; 0,-1,2,-1,0,0,0,-1; 0,0,-1,2,-1,0,0,0; 0,0,0,-1,2,-1,0,0; 0,0,0,0,-1,2,-1,0; 0,0,0,0,0,-1,2,0; 0,0,-1,0,0,0,0,2].

a(n) = 16*a(n-1)-105*a(n-2)+364*a(n-3)-714*a(n-4)+784*a(n-5)-440*a(n-6)+96*a(n-7) -a(n-8). - R. J. Mathar, May 08 2009 [Corrected by Georg Fischer, Mar 12 2020]

G.f.: -(2*x-1)*(2*x^2-4*x+1)*(x^4-16*x^3+20*x^2-8*x+1) / (1-16*x +105*x^2 -364*x^3+714*x^4-784*x^5+440*x^6-96*x^7+x^8). - R. J. Mathar, May 08 2009

Extensions

Edited by R. J. Mathar, May 08 2009