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%I #91 Mar 24 2024 02:27:46
%S 1,0,1,1,4,10,35,120,455,1792,7413,31780,140833,641928,3000361,
%T 14338702,69902535,346939792,1750071307,8958993507,46484716684,
%U 244187539270,1297395375129,6965930587924,37766629518625
%N 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 Related to triangulations of an n-gon such that all internal vertices have valence at least 6.
%C This sequence arises from the sequence G_2 polynomials in q when q is replaced by 1. The sequence of degrees of these q-polynomials (Westbury 2010) is A227849. - _Michael Somos_, Nov 01 2013
%D Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Birkhäuser Boston (2003).
%H Michael De Vlieger, <a href="/A059710/b059710.txt">Table of n, a(n) for n = 0..1200</a>
%H Georgia Benkart and A. Elduque, <a href="https://arxiv.org/abs/1606.07588">Cross products, invariants, and centralizers</a>, arXiv preprint arXiv:1606.07588 [math.RT], 2016.
%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, <a href="https://arxiv.org/abs/1911.10288">On sequences associated to the invariant theory of rank two simple Lie algebras</a>, arXiv:1911.10288 [math.CO], 2019.
%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, <a href="https://arxiv.org/abs/2110.13753">On some combinatorial sequences associated to invariant theory</a>, arXiv:2110.13753 [math.CO], 2021.
%H Juan B. Gil and Jordan O. Tirrell, <a href="https://arxiv.org/abs/1806.09065">A simple bijection for classical and enhanced k-noncrossing partitions</a>, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
%H G. Kuperberg, <a href="http://arxiv.org/abs/q-alg/9712003">Spiders for rank 2 Lie algebras</a>, arXiv:q-alg/9712003, 1997.
%H G. Kuperberg, <a href="http://projecteuclid.org/euclid.cmp/1104287237">Spiders for rank 2 Lie algebras</a>, Comm. Math. Phys. 180 (1996), 109-151.
%H Gilles Lachaud, <a href="https://doi.org/10.1090/conm/722/14536">The distribution of the trace in the compact group of type G_2</a>, in Arithmetic Geometry: Contemporary Mathematics (2019) Vol. 722, 79-103.
%H Q. Lu, W. Zheng, and Z. Zheng, <a href="http://arxiv.org/abs/1305.3405">On the distribution of Jacobi sums</a>, arXiv:1305.3405 [math.NT], 2013.
%H Robert Scherer, <a href="https://arxiv.org/abs/2003.07984">A criterion for asymptotic sharpness in the enumeration of simply generated trees</a>, arXiv:2003.07984 [math.CO], 2020.
%H Robert Scherer, <a href="https://www.math.ucdavis.edu/~tdenena/dissertations/202101_Scherer_Dissertation.pdf">Topics in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Univ. of California Davis (2021).
%H Bruce W. Westbury, <a href="http://arxiv.org/abs/math/0507112">Enumeration of non-positive planar trivalent graphs</a>, arXiv:math/0507112 [math.CO], 2005.
%H Bruce W. Westbury, <a href="http://dx.doi.org/10.1007/s10801-006-0041-4">Enumeration of non-positive planar trivalent graphs</a>, J. Algebraic Combin. 25 (2007).
%H Bruce W. Westbury, <a href="http://mathoverflow.net/questions/32020/">Finding recurrence relation for a sequence of polynomials</a> (2010).
%H Bruce W. Westbury, <a href="https://doi.org/10.37236/4569">Invariant tensors and the cyclic sieving phenomenon</a>, El. J. Combinat. 23 (4) (2016) P4.2
%F Limit_{n->oo} a(n+1)/a(n) = 7.
%F a(0)=1, a(1)=0, a(2)=1 and (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) for n > 2. - Alec Mihailovs (alec(AT)mihailovs.com), Feb 12 2005
%F Let f(n) = a(n+3)*a(n+4)*a(n+5) - 15 * a(n+4)^2*a(n+3) ... - 2744 * a(n+2)*a(n+1)*a(n), a homogeneous cubic polynomial in {a(n), a(n+1), ..., a(n+5)} with 40 terms. Then f(n) = 0 unless n = -3. - _Michael Somos_, Nov 01 2013
%F Let g(n) = 30 * a(n+3)^2*a(n+4) - 450 * a(n+3)^4 ... - 76832 * a(n+2)*a(n+1)*a(n)^2, a homogeneous quartic polynomial in {a(n), a(n+1), ..., a(n+4)} with 56 terms. Then g(n) = 0 unless n = -3. - _Michael Somos_, Nov 01 2013
%F O.g.f.: -(1-7*x)^(4/3)*(x+1)^2*(1+2*x)^(2/3)*hypergeom([-2/3, 7/3],[2],-27*x*(x+1)^2/((1+2*x)*(7*x-1)^2))/(6*x^5)+(28*x^4+66*x^3+46*x^2+15*x+1)/(6*x^5). - _Mark van Hoeij_, Jul 26 2021
%e G.f. = 1 + x^2 + x^3 + 4*x^4 + 10*x^5 + 35*x^6 + 120*x^7 + 455*x^8 + ...
%p 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:=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);
%p # See Mihailovs reference for proof that this program is correct.
%p # Alec Mihailovs, Jun 17 2003
%t a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := 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])/((n + 5)*(n + 6));
%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Nov 17 2017 *)
%o (PARI) {a(n) = if( n<1, n==0, (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)) / ((n+5)*(n+6)))}; /* _Michael Somos_, Oct 28 2013 */
%Y The analogous sequence for A_1 is A000108.
%Y See A060049 for related primitive diagrams, A227849.
%K easy,nonn
%O 0,5
%A _Greg Kuperberg_, Feb 08 2001
%E Removed "word" keyword because it is not appropriate. - Kang Seonghoon (lifthrasiir(AT)gmail.com), Oct 10 2008
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