Search: id:A059710
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%I A059710
%S A059710 1,0,1,1,4,10,35,120,455,1792,7413,31780,140833,641928,3000361,
%T A059710 14338702,69902535,346939792,1750071307,8958993507,46484716684,
%U A059710 244187539270,1297395375129,6965930587924,37766629518625
%N A059710 Dimension of space of invariants of n-th tensor power of 7-dimensional 
               irreducible representation of G_2. Also the number of n-leaf, otherwise 
               trivalent graphs in a disk such that all faces have at least 6 sides.
%C A059710 Related to triangulations of an n-gon such that all internal vertices 
               have valence at least 6.
%D A059710 G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 
               (1996), 109-151
%D A059710 Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups 
               and Algebras, Springer-Verlag New York (2003).
%H A059710 G. Kuperberg, ibid., <a href="http://front.math.ucdavis.edu/q-alg/9712003">
               arXiv:q-alg/9712003</a>
%F A059710 lim a_(n+1)/a_n = 7.
%F A059710 a(0)=1, a(1)=0, a(2)=1 and (n+5)(n+6)a(n)=2(n-1)(2n+5)a(n-1)+(n-1)(19n+18)a(n-2)+14(n-1)(n-2)a(n-3) 
               for n>2. - Alec Mihailovs (alec(AT)mihailovs.com), Feb 12 2005
%p A059710 c := x^2*y+x^3*y+x*y+x*y^2+y^2+x^3+x^4: mc := p->expand((p*c-subs(x=0,
               p*c)-subs(y=0,p*c))/x/y): g2 := proc(n) option remember; global x,
               y,c,mc; expand((mc(g2(n-1))-subs(x=0,mc(g2(n-1))))/x-subs(x=0,g2(n-1))) 
               end: g2(0) := 1: a := seq(subs(x=0,y=0,g2(n)),n=0..50);
%p A059710 A059710:=rsolve({(n+5)*(n+6)*A(n)=2*(n-1)*(2*n+5)*A(n-1)+(n-1)*(19*n+18)*A(n-2)+14*(n-1)*(n-2)*A(n-3),
               A(0)=1,A(1)=0,A(2)=1},A(n),makeproc); (Mihailovs)
%Y A059710 The analogous sequence for A_1 is A000108.
%Y A059710 Sequence in context: A030003 A149175 A149176 this_sequence A149177 A149178 
               A152916
%Y A059710 Adjacent sequences: A059707 A059708 A059709 this_sequence A059711 A059712 
               A059713
%K A059710 easy,nonn
%O A059710 0,5
%A A059710 Greg Kuperberg (greg(AT)math.ucdavis.edu), Feb 08 2001
%E A059710 Maple program from Alec Mihailovs, Jun 17 2003. See Mihailovs reference 
               for proof that program is correct.
%E A059710 Removed "word" keyword because it is not appropriate. - Kang Seonghoon 
               (lifthrasiir(AT)gmail.com), Oct 10 2008

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