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Dimensions of certain Lie algebra (see reference for precise definition).
1

%I #7 Jan 09 2024 14:16:53

%S 1,1539,617253,105489615,9743909175,561104814270,22155294211050,

%T 641798777779380,14341812253735200,256924158460700640,

%U 3803151504372077088,47661484991340720864,515805481647912249276,4900345026363722587050,41434653012551675362750,315469600749446851604325

%N Dimensions of certain Lie algebra (see reference for precise definition).

%H Paolo Xausa, <a href="/A133354/b133354.txt">Table of n, a(n) for n = 0..10000</a>

%H 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), pp. 143-179. [Th. 7.2(iii), case a = 8]

%p b:=binomial; t72c:= proc(a,k) ((4*k+3*a+2)/((3*a+2)*(k+1))) * b(k+a,k)*b(k+a+1,k)*b(k+3*a/2-1,k)*b(k+3*a/2,k)*b(2*k+2*a+1,2*k)/ (b(k+a/2-1,k)*b(k+a/2,k)*b(2*k+a,2*k)); end; [seq(t72c(8,k),k=0..40)];

%t t72c[a_,k_] := (4k+3a+2) / ((k+1)(3a+2)) Binomial[k+a,k] Binomial[k+a+1,k] Binomial[k+3/2a-1,k] Binomial[k+3/2a,k] Binomial[2k+2a+1,2k] / (Binomial[k+a/2-1,k] Binomial[k+a/2,k] Binomial[2k+a,2k]);

%t Array[t72c[8,#]&,30,0] (* _Paolo Xausa_, Jan 09 2024 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 20 2007