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

%I A128894 #32 Dec 30 2022 11:10:18
%S A128894 3,8,27,14,77,273,28,300,1925,8918,52,1053,12376,100776,627912,78,
%T A128894 2430,43758,537966,4969107,36685506,133,7371,238602,5248750,85709988,
%U A128894 1101296924,11604306012,190,15504,749360,24732110,605537790,11619550320,181746027600,2386644625950
%N A128894 Triangle read by rows, giving dimensions of exceptional groups with extension to E9 as a non-simple Lie algebra.
%C A128894 Row sums are {3, 35, 364, 11171, 742169, 42238845, 12796807780, ...}.
%H A128894 J. M. Landsberg and L. Manivel, <a href="https://doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), 143-179. [See Th. 7.1]
%H A128894 J. M. Landsberg and L. Manivel, <a href="https://hal.archives-ouvertes.fr/hal-00330636">The Sextonions and E_{7 1/2}</a>, (see p. 15), HAL Id : hal-00330636.
%H A128894 J. M. Landsberg and L. Manivel, <a href="https://arxiv.org/abs/math/0107032">Triality, exceptional Lie algebras and Deligne dimension formulas</a>, arXiv:math/0107032 [math.AG], 2001. (see page 2)
%F A128894 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.
%e A128894 Triangle begins:
%e A128894    3;
%e A128894    8,   27;
%e A128894   14,   77,   273;
%e A128894   28,  300,  1925,   8918;
%e A128894   52, 1053, 12376, 100776,  627912;
%e A128894   78, 2430, 43758, 537966, 4969107, 36685506; ...
%t A128894 p = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16};
%t A128894 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]);
%t A128894 Table[Table[g[p[[n]], k], {k, 1, n}], {n, 1, Length[p]}]
%Y A128894 Cf. A133238.
%K A128894 nonn,tabl
%O A128894 1,1
%A A128894 _Roger L. Bagula_, May 09 2007