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].
1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867145, 3269034, 12446307, 47767466, 184508963, 716386598, 2793067210, 10926148172, 42857189054, 168471757292, 663434825367, 2616336659586, 10329939578230
Offset: 0
Examples
a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0].
Links
- Wikipedia, E_7 (mathematics).
- Index entries for linear recurrences with constant coefficients, signature (12,-54,112,-105,36,-1).
Programs
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Maple
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
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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
Formula
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
Extensions
More terms from Paul D. Hanna, Jan 02 2007