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.

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

Offset: 0

Author

Greg Kuperberg, Feb 08 2001

Keywords

• easy
• nonn

Comments

Related to triangulations of an n-gon such that all internal vertices have valence at least 6.

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

Examples

			G.f. = 1 + x^2 + x^3 + 4*x^4 + 10*x^5 + 35*x^6 + 120*x^7 + 455*x^8 + ...
		

References

• Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Birkhäuser Boston (2003).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1200
• Georgia Benkart and A. Elduque, Cross products, invariants, and centralizers, arXiv preprint arXiv:1606.07588 [math.RT], 2016.
• Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
• Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
• Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
• G. Kuperberg, Spiders for rank 2 Lie algebras, arXiv:q-alg/9712003, 1997.
• G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151.
• Gilles Lachaud, The distribution of the trace in the compact group of type G_2, in Arithmetic Geometry: Contemporary Mathematics (2019) Vol. 722, 79-103.
• Q. Lu, W. Zheng, and Z. Zheng, On the distribution of Jacobi sums, arXiv:1305.3405 [math.NT], 2013.
• Robert Scherer, A criterion for asymptotic sharpness in the enumeration of simply generated trees, arXiv:2003.07984 [math.CO], 2020.
• Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021).
• Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, arXiv:math/0507112 [math.CO], 2005.
• Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007).
• Bruce W. Westbury, Finding recurrence relation for a sequence of polynomials (2010).
• Bruce W. Westbury, Invariant tensors and the cyclic sieving phenomenon, El. J. Combinat. 23 (4) (2016) P4.2

Crossrefs

The analogous sequence for A_1 is A000108.

See A060049 for related primitive diagrams, A227849.

Programs

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);
# See Mihailovs reference for proof that this program is correct.
# Alec Mihailovs, Jun 17 2003

Mathematica

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));
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 17 2017 *)

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 */

Formula

Limit_{n->oo} 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)*(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

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

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

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

Extensions

Removed "word" keyword because it is not appropriate. - Kang Seonghoon (lifthrasiir(AT)gmail.com), Oct 10 2008