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A128894
Triangle read by rows, giving dimensions of exceptional groups with extension to E9 as a non-simple Lie algebra.
3
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
1,1
COMMENTS
Row sums are {3, 35, 364, 11171, 742169, 42238845, 12796807780, ...}.
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)
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.
EXAMPLE
Triangle begins:
3;
8, 27;
14, 77, 273;
28, 300, 1925, 8918;
52, 1053, 12376, 100776, 627912;
78, 2430, 43758, 537966, 4969107, 36685506; ...
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]}]
CROSSREFS
Cf. A133238.
Sequence in context: A298624 A088590 A023637 * A118165 A296258 A066020
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, May 09 2007
STATUS
approved