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A133238 Dimensions of certain Lie algebra (see reference for precise definition).

%I A133238 #16 Jan 11 2024 10:59:49
%S A133238 1,52,1053,12376,100776,627912,3187041,13748020,51949755,175847880,
%T A133238 542393670,1544927904,4107092288,10278624864,24388573014,55188666312,
%U A133238 119696471453,249869263644,503865726155,984563860280,1869304764600,3456658569000,6238533257775
%N A133238 Dimensions of certain Lie algebra (see reference for precise definition).
%H A133238 Paolo Xausa, <a href="/A133238/b133238.txt">Table of n, a(n) for n = 0..10000</a>
%H A133238 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. [Th. 7.1, case a=1]
%F A133238 Empirical g.f.: (x^8+36*x^7+341*x^6+1208*x^5+1820*x^4+1208*x^3+341*x^2+36*x+1) / (x-1)^16. - _Colin Barker_, Jul 27 2013
%p A133238 b:=binomial; t71:= proc(a,k) ((3*a+2*k+5)/(3*a+5)) * b(k+2*a+3,k)*b(k+5*a/2+3,k)*b(k+3*a+4,k)/(b(k+a/2+1,k)*b(k+a+1,k)); end; [seq(t71(1,k),k=0..30)];
%t A133238 t71[a_, k_] := (3a+2k+5) / (3a+5) Binomial[k+2a+3,k] Binomial[k+5/2a+3,k] Binomial[k+3a+4,k] / (Binomial[k+a/2+1,k] Binomial[k+a+1,k]);
%t A133238 Array[t71[1,#]&,30,0] (* _Paolo Xausa_, Jan 11 2024 *)
%Y A133238 The cases a = -4/3, -1, -2/3, 0, 1, 2, 4, 6, 8 of Th. 7.1 of Landsberg and Manivel give sequences A005408, A000578, A085462, A107942, A133238 (this entry), A133239, A133240, A133241 and A030650 respectively. See also triangle in A128894.
%K A133238 nonn
%O A133238 0,2
%A A133238 _N. J. A. Sloane_, Oct 15 2007