A117133 Dimensions of the traditional Cartan exceptional group sequence A1, G2, F4, E6, E7, E8.

3, 14, 52, 78, 133, 248

Offset: 1

Author

Roger L. Bagula, May 17 2007

Keywords

• nonn
• fini
• full
• uned

Crossrefs

Cf. A129766.

Programs

Mathematica

a[0] = {1};
a[1] = {1, 5};
a[2] = {1, 5, 7, 11};
a[3] = {1, 4, 5, 7, 8, 11};
a[4] = {1, 5, 7, 9, 11, 13, 17};
a[5] = {1, 7, 11, 13, 17, 19, 23, 29};
b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}]

Formula

P(n) = Poincare-Polynomial(n) = Product_{m=1..n} (1 + t^A129766(m)); a(n) = Length(CoefficientList(P(n),t)) - 1.