A059710 -id:A059710 - cp's OEIS frontend

A095922 Dimension of invariants of n-th tensor power of 5-dimensional irreducible representation of B_2.

1, 0, 1, 0, 3, 1, 15, 15, 105, 190, 945, 2410, 10263, 31890, 127699, 444458, 1751685, 6518736, 25807445, 100152288, 401449271, 1602902055, 6519160851, 26580508625, 109656966853, 454524861846, 1899821492925, 7982263725826, 33757439931675

Offset: 0

Alec Mihailovs (alec(AT)mihailovs.com), Jul 11 2004

The analogous sequence for G_2 is A059710.

			a(2)=1 because SO(5) has unique (up to multiplication by a constant) invariant in V ⊗ V - the quadratic form x^2+y^2+z^2+u^2+v^2.

Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Springer-Verlag New York (2004).

T. D. Noe, Table of n, a(n) for n = 0..500
G. Lachaud, On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius, arXiv preprint arXiv:1506.06482, 2015.

Cf. A000108, A059710, A247304.

ca:=n->binomial(n+n,n)/(n+1); a:=n->add(ca(i)*ca(i+1)*binomial(n,2*i),i=0..floor(n/2))- add(ca(i)^2*binomial(n,2*i-1),i=0..floor((n+1)/2)); seq(a(n),n=0..40);
A095922:=rsolve({(n+3)*(n+4)*A(n)=3*(n-1)*(n+2)*A(n-1)+(n-1)*(13*n+4)*A(n-2)-15*(n-1)*(n-2)*A(n-3),A(0)=1,A(1)=0,A(2)=1},A(n),makeproc);

t = {0, 1, 0}; Do[AppendTo[ t, (3 (n - 1) (n + 2) t[[n - 1]] + (n - 1) (13 n + 4) t[[n - 2]] - 15 (n - 1) (n - 2) t[[n - 3]])/((n + 3) (n + 4))], {n, 4, 25}]; t = Join[{1}, t] (* T. D. Noe, Apr 11 2014 *)
a[n_] := -n*HypergeometricPFQ[{3/2, 1/2 - n/2, 1 - n/2}, {3, 3}, 16] + HypergeometricPFQ[{3/2, 1/2 - n/2, -n/2}, {2, 3}, 16]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 03 2016 *)

a(n) =sum(A000108(i)*A000108(i+1)*binomial(n, 2*i), i=0..floor(n/2)) - sum(A000108(i)^2*binomial(n, 2*i-1), i=0..floor((n+1)/2)); exponential generating function = exp(t)*b(t) where b(t) is the exponential generating function of the sequence B(n) = (-1)^n*A000108(floor((n+1)/2))*A000108(floor(n/2+1)).
a(0)=1, a(1)=0, a(2)=1 and (n+3)(n+4)a(n)=3(n-1)(n+2)a(n-1)+(n-1)(13n+4)a(n-2)-15(n-1)(n-2)a(n-3) for n>2.
a(n) ~ 3 * 5^(n+5) / (128*Pi*n^5). - Vaclav Kotesovec, Oct 03 2016

A060049 Triangulations of an n-gon such that each internal vertex has valence at least 6, i.e., nonpositively curved triangulations.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 15, 50, 181, 697, 2821, 11892, 51874, 232974, 1073070, 5053029, 24264565, 118570292, 588567257, 2963358162, 15114174106, 78004013763, 406971280545, 2144659072330, 11407141925639, 61197287846831

Offset: 0

Greg Kuperberg, Feb 15 2001

This is the connected version of A059710 in the following sense. Let C(x) be the ordinary generating function for this sequence and A(x) the ordinary generating function for A059710. Then these satisfy the functional equation A(x) = C(x*A(x)). - Bruce Westbury, Nov 05 2013

			a(6) = 15 because there are 14 = A000108(4) triangulations without internal vertices, plus the triangulation with 6 pie slices.

Bruce Westbury, Table of n, a(n) for n = 0..39
Greg Kuperberg, Spiders for rank 2 Lie algebras, arXiv:q-alg/9712003, 1997.
Greg Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151.
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)

Cf. A059710.

The g.f. B(x) is derived from the g.f. A(x) of A059710 by A(x) = A(x*B(x))+1.

A227849 a(n) = 2 * floor( 3/14 * n^2) if n even, a(n) = 2 * round( 3/14 * n^2) -1 if n odd.

Original entry on oeis.org

0, -1, 0, 3, 6, 9, 14, 21, 26, 33, 42, 51, 60, 71, 84, 95, 108, 123, 138, 153, 170, 189, 206, 225, 246, 267, 288, 311, 336, 359, 384, 411, 438, 465, 494, 525, 554, 585, 618, 651, 684, 719, 756, 791, 828, 867, 906, 945, 986, 1029, 1070, 1113, 1158, 1203, 1248

Offset: 0

Michael Somos, Oct 31 2013

sign

The degrees of the sequence of the G_2 polynomials defined by Bruce Westbury is conjectured to be a(n).

			G.f. = -x + 3*x^3 + 6*x^4 + 9*x^5 + 14*x^6 + 21*x^7 + 26*x^8 + 33*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
B. W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007), 357-373, arXiv:math/0507112 [math.CO], 2005.
B. W. Westbury, Finding recurrence relation for a sequence of polynomials, MathOverflow, 15 July 2010.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 1, -2, 1).

Cf. A059710.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^8-2*x^7-2*x^6-2*x^3-2*x^2+x)/(x^9-2*x^8+x^7-x^2+2*x-1))); // G. C. Greubel, Aug 08 2018

CoefficientList[Series[(x^8-2*x^7-2*x^6-2*x^3-2*x^2+x)/(x^9-2*x^8+x^7 - x^2+2*x-1), {x, 0, 50}], x] (* G. C. Greubel, Aug 08 2018 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,-1,0,3,6,9,14,21,26},60] (* Harvey P. Dale, Jul 26 2022 *)

{a(n) = (n%2*7 + 3*n^2) \ 14 * 2 - n%2}

{a(n) = (3*n^2 - [0, 10, 12, 6, 6, 12, 10][n%7 + 1]) / 7}

G.f.: (x^8 - 2*x^7 - 2*x^6 - 2*x^3 - 2*x^2 + x) / (x^9 - 2*x^8 + x^7 - x^2 + 2*x - 1).
G.f.: -x * (1 + x) * (1 - 3*x + x^2 - x^3 + x^4 - 3*x^5 + x^6) / ((1 - x)^2 * (1 - x^7)).
a(-n) = a(n). a(n+7) = a(n) + 3*(2*n + 7).

A060050 Number of irreducible nonpositively curved triangulations of an n-gon: All internal vertices have at valence at least 6 and no diagonals of the n-gon are allowed.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 5, 13, 46, 155, 561, 2068, 7871, 30586, 121391, 490196, 2011422, 8370698, 35285987, 150485667, 648653910, 2823402675, 12400659846, 54920758496, 245126368841, 1101983749921, 4987538210079, 22716326086134

Offset: 2

Greg Kuperberg, Feb 15 2001

			c(8) = 5 = 1+4. We can divide the octagon into 8 pie slices and we can split any pair of opposite radii of this triangulation into two triangles.

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.

Cf. A060049, A059710.

The g.f. C(x) is derived from the g.f. B(x) of A060049 by B_1(x) = C_1(B_1(x))+x, where B_1(x) = B(x)/x and C_1(x) = C(x)/x.

A247591 Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.

Original entry on oeis.org

1, 1, 3, 16, 126, 1296, 16071, 228514, 3607890, 61891050, 1135871490, 22049362440, 448790912004, 9512960347260, 208858963314735, 4728736078065810, 110006925920592810, 2621619942885055530, 63840054782606886630, 1585094577104979776880, 40054740803371374834780, 1028483346608802276173280

Offset: 0

Paul Zinn-Justin, Sep 20 2014

nonn

The 6-dimensional representation is the usual representation of SO(6)=A_3.

			For 2n=6, there are 15 invariants corresponding to all ways of pairing the 6 indices with the metric tensor, plus one invariant which is the completely skew-symmetric 6-index tensor.

Cf. A095922, A059710.

a[0] = 1; a[1] = 1; a[n_] := a[n] = (4*n*(2*n-1)*(5*n+7)*a[n-1] - 36*(n-1)*(2*n-3)*(2*n-1)*a[n-2]) / ((n+2)*(n+3)^2); Table[a[n], {n, 0, 21}]

N=66; v=vector(N); v[1]=1; v[2]=1;
for(n=2, N-1, my(t=n+1); v[t] = (-36*(n-1)*(2*n-3)*(2*n-1)*v[t-2] + 4*n*(2*n-1)*(5*n+7)*v[t-1]) / ((n+2)*(n+3)^2) );
v \\ Joerg Arndt, Sep 20 2014

a(n) = (-36*(n-1)*(2*n-3)*(2*n-1)*a(n-2) + 4*n*(2*n-1)*(5*n+7)*a(n-1)) / ((n+2)*(n+3)^2).

a(n) = (9*(n+1)*A005802(n)-(n+5)*A005802(n+1))*binomial(2*n,n)/(2*(n+1)*(n+2)). - Mark van Hoeij, Nov 12 2023

A353173 Dimension of space of invariants of n-th tensor power of the 26-dimensional fundamental (or "standard") irreducible representation of F_4.

Original entry on oeis.org

1, 0, 1, 1, 5, 15, 70, 330, 1820, 10858, 70875, 497135, 3727955, 29658410, 248989676, 2194891440, 20231692430, 194286848280, 1937546532820, 20008993160460, 213436182918652, 2346406693816315, 26531060178217182, 307987244037724262, 3664579007885995952

Offset: 0

David A. Madore, Apr 28 2022

nonn

It is known that a(n) satisfies a linear recurrence relation with polynomial coefficients. The limit of a(n+1)/a(n) is 26.

			a(1)=0 because there is no F_4-invariant linear form on the 26-dimensional representation; a(2)=1 because there is, up to scalars, precisely one invariant quadratic form.

MathOverflow, Invariants for the exceptional complex simple Lie algebra F4

The analogous sequence for the (52-dimensional) adjoint representation of F_4 is: A179685.
A similar sequence for G_2 (for its 7-dimensional fundamental irreducible representation) is: A059710.
A similar sequence for B_2 (for its standard 5-dimensional irreducible representation) is: A095922.
For A_n the similar sequence (omitting some 0's) is given by the (n+1)-dimensional Catalan numbers, e.g., A005789 for A_2.

p_tensor(n,[0,0,0,1],F4)|[0,0,0,0] # Returns the value of a(n).

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A095922 Dimension of invariants of n-th tensor power of 5-dimensional irreducible representation of B_2.

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A060049 Triangulations of an n-gon such that each internal vertex has valence at least 6, i.e., nonpositively curved triangulations.

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A227849 a(n) = 2 * floor( 3/14 * n^2) if n even, a(n) = 2 * round( 3/14 * n^2) -1 if n odd.

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A060050 Number of irreducible nonpositively curved triangulations of an n-gon: All internal vertices have at valence at least 6 and no diagonals of the n-gon are allowed.

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A247591 Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.

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A353173 Dimension of space of invariants of n-th tensor power of the 26-dimensional fundamental (or "standard") irreducible representation of F_4.

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