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%I #10 Mar 12 2020 10:27:59
%S 1,2,5,14,42,132,430,1444,4981,17594,63442,232828,867146,3269060,
%T 12446684,47771496,184544427,716658870,2794956099,10938266562,
%U 42930256917,168890693650,665739119129,2628578437646,10393091551794,41141896235012,163028816478833
%N Top-left "head" entry of the n-th power of the E8 Cartan matrix.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/E8_(Mathematics)">E8</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (16,-105,364,-714,784,-440,96,-1).
%F 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].
%F 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]
%F 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
%e a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0,0].
%p 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 ] ]) ;
%p 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
%Y Cf. A126566, A126567, A126568.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Dec 28 2006
%E Edited by _R. J. Mathar_, May 08 2009
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