A143076 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).

-1, 0, -274, 3480, -96281, 1983736, -44477455, 973668972, -21447199320, 471699618464, -10378093042737, 228314605056428, -5022937786817620, 110504554075400128, -2431100545346633371, 53484210384313253132, -1176652635677112260460

Offset

1,3

Comments

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

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

Links

Table of n, a(n) for n=1..17.

Formula

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.

Mathematica

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}]

Crossrefs

Sequence in context: A322254 A204278 A165677 * A171346 A001242 A214111

Adjacent sequences: A143073 A143074 A143075 * A143077 A143078 A143079

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 14 2008

Status

approved