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A030648 Dimensions of multiples of minimal representation of complex Lie algebra E6.

%I A030648 #29 Feb 10 2025 09:32:29
%S A030648 1,27,351,3003,19305,100386,442442,1706562,5895396,18559580,53965548,
%T A030648 146477916,374332452,907036326,2096092350,4642456390,9895762305,
%U A030648 20373628275,40639459575,78751105875,148599912825,273612537900,492502568220,868056366060,1500344336400
%N A030648 Dimensions of multiples of minimal representation of complex Lie algebra E6.
%D A030648 Onishchik and Vinberg, Seminar on Lie Groups and Algebraic Groups, Springer Verlag 1990, see Table 5.
%H A030648 T. D. Noe, <a href="/A030648/b030648.txt">Table of n, a(n) for n = 0..1000</a>
%H A030648 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. 7.3, case a=8]
%H A030648 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).
%F A030648 a(n) = (1/517440)*binomial(n+11, 3)*binomial(n+3, 3)*binomial(n+8, 5)^2.
%F A030648 From _G. C. Greubel_, Feb 09 2025: (Start)
%F A030648 a(n) = (1/165)*binomial(n+8,8)*binomial(n+11,8).
%F A030648 G.f.: (1 + 10*x + 28*x^2 + 28*x^3 + 10*x^4 + x^5)/(1-x)^17. (End)
%p A030648 b:=binomial; t73:= proc(a,k) ((2*k+a)*(k+a)/(a^2)) * b(k+a-1,k)*b(k+3*a/2-1,k)/(b(k+a/2,k)); end; [seq(t73(8,k),k=0..40)];
%t A030648 Table[(Binomial[n+11,3]Binomial[n+3,3]Binomial[n+8,5]^2)/517440,{n,0,30}]  (* _Harvey P. Dale_, May 01 2011 *)
%o A030648 (Magma)
%o A030648 A030648:= func< n | Binomial(n+8,8)*Binomial(n+11,8)/165 >;
%o A030648 [A030648(n): n in [0..30]]; // _G. C. Greubel_, Feb 09 2025
%o A030648 (SageMath)
%o A030648 def A030648(n): return binomial(n+8,8)*binomial(n+11,8)//165
%o A030648 print([A030648(n) for n in range(31)]) # _G. C. Greubel_, Feb 09 2025
%Y A030648 Cf. A133355.
%K A030648 nonn,easy
%O A030648 0,2
%A A030648 Paolo Dominici (pl.dm(AT)libero.it)
%E A030648 Edited by _N. J. A. Sloane_, Oct 20 2007