A133350 - OEIS

%I #19 Jan 09 2024 08:45:00

%S 1,90,1274,8568,38115,130130,369460,915552,2043621,4198810,8065134,

%T 14651000,25393095,42280434,68000360,106108288,161222985,239249178,

%U 347629282,495626040,694637867,958548690,1304114076,1751385440,2324174125,3050557146,3963426390

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

%H Paolo Xausa, <a href="/A133350/b133350.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 = 1]

%F Empirical g.f.: (14*x^5+273*x^4+840*x^3+582*x^2+82*x+1) / (x-1)^8. - _Colin Barker_, Jul 27 2013

%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(1,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[1,#]&,30,0] (* _Paolo Xausa_, Jan 09 2024 *)

%K nonn

%O 0,2

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