%I #3 Oct 12 2012 14:54:52

%S -1,0,-274,3480,-96281,1983736,-44477455,973668972,-21447199320,

%T 471699618464,-10378093042737,228314605056428,-5022937786817620,

%U 110504554075400128,-2431100545346633371,53484210384313253132,-1176652635677112260460

%N Sequence from expansion of Cartan E_11 12 state root sum zero characteristic polynomial: p(x)=1/(-1 + 274 x^2 - 3480 x^3 + 21205 x^4 - 76696 x^5 + 175891x^6 - 259324 x^7 + 240551 x^8 - 131824 x^9 + 37101 x^10 - 3676 x^11 - 44 x^12).

%C The root that balances the Cartan matrices characteristic polynomial roots is: x=-Trace[Cartan_Matrix];

%C Sum[x /. NSolve[p[x] == 0, x][[n]], {n, 1, 12}]=-3.552713678800501*10^(-15).

%F p(x)=1/(-1 + 274 x^2 - 3480 x^3 + 21205 x^4 - 76696 x^5 + 175891x^6 - 259324 x^7 + 240551 x^8 - 131824 x^9 + 37101 x^10 - 3676 x^11 - 44 x^12); p(x)=Sum[a(n)*x^n,{n,0,Infinity}]; a(n) output.

%t Clear[m11, p, f]; m11 = {{2, -1, 0,0, 0, 0, 0,0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -1, 2, -1, 0, 0, 0,0, 0, 0, -1}, {0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, -1, 2, -1,0, 0, 0, 0, 0}, {0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0}, {0, 0, 0,0, 0, 0, 0, -1, 2, -1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2}}; p[x_] = ExpandAll[(x + Sum[m11[[n, n]], {n, 1, Length[m11]}])*CharacteristicPolynomial[m11, x]]; f[x_] = ExpandAll[1/(x^12*p[1/x])]; a = Table[SeriesCoefficient[Series[f[t], {t, 0, 35}], n], {n, 0, 35}]

%K uned,sign

%O 1,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 14 2008