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A125501 The (1,1)-entry in the matrix M^n, where M is the 7 X 7 Cartan matrix [2,-1,0,0,0,0,0; -1,2,-1,0,0,0,0; 0,-1,2,-1,0,0,-1; 0,0,-1,2,-1,0,0; 0,0,0,-1,2,-1,0; 0,0,0,0,-1,2,0; 0,0,-1,0,0,0,2].

%I A125501 #19 May 03 2024 19:55:29
%S A125501 1,2,5,14,42,132,430,1444,4981,17594,63442,232828,867145,3269034,
%T A125501 12446307,47767466,184508963,716386598,2793067210,10926148172,
%U A125501 42857189054,168471757292,663434825367,2616336659586,10329939578230
%N A125501 The (1,1)-entry in the matrix M^n, where M is the 7 X 7 Cartan matrix [2,-1,0,0,0,0,0; -1,2,-1,0,0,0,0; 0,-1,2,-1,0,0,-1; 0,0,-1,2,-1,0,0; 0,0,0,-1,2,-1,0; 0,0,0,0,-1,2,0; 0,0,-1,0,0,0,2].
%H A125501 Wikipedia, <a href="https://en.wikipedia.org/wiki/E7_(mathematics)">E_7 (mathematics)</a>.
%H A125501 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,112,-105,36,-1).
%F A125501 G.f.: -(2*x-1)*(x^4 - 12*x^3 + 19*x^2 - 8*x + 1) / (x^6 - 36*x^5 + 105*x^4 - 112*x^3 + 54*x^2 - 12*x + 1). - _Colin Barker_, May 25 2013
%e A125501 a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0].
%p A125501 with(linalg): M[1]:=matrix(7,7,[2,-1,0,0,0,0,0,-1,2,-1,0,0,0,0,0,-1,2,-1,0,0,-1,0,0,-1,2,-1,0,0,0,0,0,-1,2,-1,0,0,0,0,0,-1,2,0,0,0,-1,0,0,0,2]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od:1, seq(M[n][1,1],n=1..30); # _Emeric Deutsch_, Jan 20 2007
%o A125501 (PARI) {a(n)=local(E7=[2,-1,0,0,0,0,0; -1,2,-1,0,0,0,0; 0,-1,2,-1,0,0,-1; 0,0,-1,2,-1,0,0; 0,0,0,-1,2,-1,0; 0,0,0,0,-1,2,0; 0,0,-1,0,0,0,2]); (E7^n)[1,1]} \\ _Paul D. Hanna_, Jan 02 2007
%Y A125501 Cf. A126566, A126567, A126569.
%K A125501 nonn,easy
%O A125501 0,2
%A A125501 _Gary W. Adamson_, Dec 28 2006
%E A125501 More terms from _Paul D. Hanna_, Jan 02 2007