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A059710
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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.
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4
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1, 0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 31780, 140833, 641928, 3000361, 14338702, 69902535, 346939792, 1750071307, 8958993507, 46484716684, 244187539270, 1297395375129, 6965930587924, 37766629518625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Related to triangulations of an n-gon such that all internal vertices have valence at least 6.
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REFERENCES
| G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151
Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Springer-Verlag New York (2003).
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LINKS
| G. Kuperberg, ibid., arXiv:q-alg/9712003
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FORMULA
| lim a_(n+1)/a_n = 7.
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
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MAPLE
| 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);
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)
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CROSSREFS
| The analogous sequence for A_1 is A000108.
Sequence in context: A030003 A149175 A149176 * A149177 A149178 A152916
Adjacent sequences: A059707 A059708 A059709 * A059711 A059712 A059713
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KEYWORD
| easy,nonn
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AUTHOR
| Greg Kuperberg (greg(AT)math.ucdavis.edu), Feb 08 2001
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EXTENSIONS
| Maple program from Alec Mihailovs, Jun 17 2003. See Mihailovs reference for proof that program is correct.
Removed "word" keyword because it is not appropriate. - Kang Seonghoon (lifthrasiir(AT)gmail.com), Oct 10 2008
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