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A126569 Top-left "head" entry of the n-th power of the E8 Cartan matrix. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..26.

Wikipedia, E8

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) = 16a(n-1)-105a(n-2)+364a(n-3)-714a(n-4)+784(n-5)-440a(n-6)+96a(n-7) -a(n-8). 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

EXAMPLE

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

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

CROSSREFS

Cf. A126566, A126567, A126568.

Sequence in context: A057413 A126567 A125501 * A162748 A061815 A202061

Adjacent sequences:  A126566 A126567 A126568 * A126570 A126571 A126572

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Dec 28 2006

EXTENSIONS

Edited by R. J. Mathar, May 08 2009

STATUS

approved

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Last modified October 18 12:24 EDT 2018. Contains 316321 sequences. (Running on oeis4.)