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

0,2

Links

Table of n, a(n) for n=0..24.

Wikipedia, E_7 (mathematics).

Index entries for linear recurrences with constant coefficients, signature (12,-54,112,-105,36,-1).

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

Example

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

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

Prog

(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

Crossrefs

Cf. A126566, A126567, A126569.

Sequence in context: A000108 A057413 A126567 * A126569 A162748 A061815

Adjacent sequences: A125498 A125499 A125500 * A125502 A125503 A125504

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 28 2006

Extensions

More terms from Paul D. Hanna, Jan 02 2007

Status

approved