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A133317 Dimensions of certain Lie algebra (see reference for precise definition).

%I A133317 #17 Jan 10 2024 16:31:39
%S A133317 1,35,405,2695,12740,47628,149940,413820,1029105,2351635,5010005,
%T A133317 10061415,19211920,35119280,61799760,105163632,173707785,279397755,
%U A133317 438775645,674334815,1016206884,1504211500,2190324500,3141625500,4443791625,6205210011,8561787885
%N A133317 Dimensions of certain Lie algebra (see reference for precise definition).
%C A133317 This is the case P(5,n) of the family of sequences defined in A132458. - Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007
%H A133317 Paolo Xausa, <a href="/A133317/b133317.txt">Table of n, a(n) for n = 0..10000</a>
%H A133317 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. 7.2(i), case a=2]
%F A133317 Empirical g.f.: (x+1)*(x^4+24*x^3+76*x^2+24*x+1) / (x-1)^10. - _Colin Barker_, Jul 27 2013
%p A133317 b:=binomial; t72a:= proc(a,k) ((2*a+2*k+1)/(2*a+1)) * b(k+3*a/2-1,k)*b(k+3*a/2+1,k)*b(k+2*a,k)/(b(k+a/2-1,k)*b(k+a/2+1,k)); end; [seq(t72a(2,k),k=0..40)];
%t A133317 t72a[a_, k_] := (2k+2a+1) / (2a+1) Binomial[k+3/2a-1, k] Binomial[k+3/2a+1, k] Binomial[k+2a,k] / (Binomial[k+a/2-1, k] Binomial[k+a/2+1, k]);
%t A133317 Array[t72a[2, #]&, 30, 0] (* _Paolo Xausa_, Jan 10 2024 *)
%K A133317 nonn
%O A133317 0,2
%A A133317 _N. J. A. Sloane_, Oct 19 2007