A128894 Triangle read by rows, giving dimensions of exceptional groups with extension to E9 as a non-simple Lie algebra.

3, 8, 27, 14, 77, 273, 28, 300, 1925, 8918, 52, 1053, 12376, 100776, 627912, 78, 2430, 43758, 537966, 4969107, 36685506, 133, 7371, 238602, 5248750, 85709988, 1101296924, 11604306012, 190, 15504, 749360, 24732110, 605537790, 11619550320, 181746027600, 2386644625950

Offset: 1

Author

Roger L. Bagula, May 09 2007

Keywords

• nonn
• tabl

Comments

Row sums are {3, 35, 364, 11171, 742169, 42238845, 12796807780, ...}.

Examples

			Triangle begins:
   3;
   8,   27;
  14,   77,   273;
  28,  300,  1925,   8918;
  52, 1053, 12376, 100776,  627912;
  78, 2430, 43758, 537966, 4969107, 36685506; ...
		

Links

• J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [See Th. 7.1]
• J. M. Landsberg and L. Manivel, The Sextonions and E_{7 1/2}, (see p. 15), HAL Id : hal-00330636.
• J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, arXiv:math/0107032 [math.AG], 2001. (see page 2)

Crossrefs

Cf. A133238.

Programs

Mathematica

p = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16};
g[p_, k_] := (3*p +2*k +5) *Binomial[k+2*p+3, k]*Binomial[k+5*p/2 +3, k]*Binomial[k+3*p+4, k]/((3*p + 5)*Binomial[k+p/2 +1, k]*Binomial[k+p+1, k]);
Table[Table[g[p[[n]], k], {k, 1, n}], {n, 1, Length[p]}]

Formula

Let p = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16} then g(p,k) = (3*p + 2*k + 5)*binomial(k + 2*p + 3, k)*binomial(k + 5*p/2 + 3, k)*binomial(k + 3*p + 4, k)/((3*p + 5)*binomial(k + p/2 + 1, k)*binomial(k + p + 1, k)); see the Mathematica program.