A247304 -id:A247304 - cp's OEIS frontend

A099251 Bisection of Motzkin sums (A005043).

1, 1, 3, 15, 91, 603, 4213, 30537, 227475, 1730787, 13393689, 105089229, 834086421, 6684761125, 54022715451, 439742222071, 3602118427251, 29671013856627, 245613376802185, 2042162142208813, 17047255430494497, 142816973618414817

Offset: 0

N. J. A. Sloane, Nov 16 2004

The Kn4 triangle sums of A175136 lead to the sequence given above (n >= 1). For the definition of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 06 2011
Equals the expected value of trace(O)^(2n), where O is a 3 X 3 orthogonal matrix randomly selected according to Haar measure (see MathOverflow link). - Nathaniel Johnston, Sep 05 2014
From Petros Hadjicostas, Jul 23 2020: (Start)
In Smith (1985), we apparently have a(n) = P(2*n), where P(n) is the number of linearly independent three-dimensional n-th order isotropic tensors. In the paper, he refers to Smith (1968) for more details. It is not clear why he does not list the values of P(2*n+1). See also the 1978 letter of D. L. Andrews to N. J. A. Sloane.
Eric Weisstein gives some details on how the material in Smith (1968) about isotropic tensors is related to Motzkin sums. (End)

G. F. Smith, On isotropic tensors and rotation tensors of dimension m and order n, Tensor (N.S.), Vol. 19 (1968), 79-88 (MR0224008).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
Georgia Benkart and A. Elduque, Cross products, invariants, and centralizers, arXiv:1606.07588 [math.RT], 2016.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
MathOverflow, Moments of the trace of orthogonal matrices.
G. F. Smith, Lectures on constitutive expressions, Mathematical models and methods in mechanics, pp. 645-678, Banach Center Publ., 15, PWN, Warsaw, 1985 (MR0874855). See p. 653.
Eric Weisstein's World of Mathematics, Isotropic tensor.

Cf. A005043, A099252, A246860, A247304.

G := (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)): Gser := series(G,x=0,60):
1, seq(coeff(Gser, x^(2*n)), n=1..25); # Emeric Deutsch
a := n -> hypergeom([1/2, -2*n], [2], 4):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Jul 25 2020

Take[CoefficientList[Series[(1 + x - Sqrt[1 - 2 * x - 3 * x^2])/(2 * x * (1 + x)), {x, 0, 60}], x], {1, -1, 2}] (* Vaclav Kotesovec, Oct 17 2012 *)

a(n):=sum(binomial(2*j,j)*(-1)^(j)*binomial(2*n+1,j+1),j,0,2*n+1)/(2*n+1); /*Vladimir Kruchinin, Apr 02 2017*/

x='x+O('x^66); v=Vec((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, May 12 2013

Recurrence: n*(2*n + 1)*a(n) = (2*n - 1)*(13*n - 10)*a(n-1) - 3*(26*n^2 - 87*n + 76)*a(n-2) + 27*(n - 2)*(2*n - 5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n + 3/2)/(16*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
Conjecture: a(n) = (2/Pi)*Integral_{t=0..1} sqrt((1 - t)/t)*(1 - 8*t + 16*t^2)^n. - Benedict W. J. Irwin, Oct 05 2016
a(n) = Sum_{j=0..2*n+1} (C(2*j,j)*(-1)^(j)*C(2*n+1,j+1))/(2*n+1). - Vladimir Kruchinin, Apr 02 2017
a(n) = hypergeom([1/2, -2*n], [2], 4). - Peter Luschny, Jul 25 2020

More terms from Emeric Deutsch, Nov 18 2004

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

Original entry on oeis.org

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

A246860 Expected value of trace(O)^(2n), where O is a 4 X 4 orthogonal matrix randomly selected according to Haar measure.

Original entry on oeis.org

1, 3, 15, 105, 903, 8778, 92235, 1023165, 11821953, 141061206, 1727926291, 21634600078, 275950576450, 3576315994020, 46995014634435, 625082431593285, 8403885851894445, 114069364107664350, 1561609592248119645, 21543838447412548410, 299299110959202973710

Offset: 1

Nathaniel Johnston, Sep 05 2014

nonn

The corresponding sequences for 2 X 2, 3 X 3, and 5 X 5 matrices are A001700, A099251, and A247304.

a(n) is the number of triangulations with middle chord of an 2n+2-gon modulo the cyclic action. So a(n) = A000108(n)^2 - A000107(A000108(n)-1). The first part A000108(n)^2 means the cartes of two n+2-gons separated by the middle chord, second part is the duplicated joins need to be removed. - Yuchun Ji, Aug 11 2020

MathOverflow, Moments of the trace of orthogonal matrices

Cf. A000108, A001700, A099251, A247304, A247306.

A246860 := proc (n) return (1/8)*integrate(integrate((cos(x)-cos(y))^2*(2*cos(x)+2*cos(y))^(2*n), y = 0 .. 2*Pi), x = 0 .. 2*Pi)/Pi^2+(1/2)*integrate((1-cos(z)^2)*(2*cos(z))^(2*n), z = 0 .. 2*Pi)/Pi end proc; seq(A246860(n), n = 1 .. 21);

a[n_] := a[n] = (1/8)*Integrate[Integrate[(Cos[x] - Cos[y])^2 * (2 Cos[x] + 2 Cos[y])^(2 n), {y, 0, 2 Pi}], {x, 0, 2 Pi}]/ Pi^2 + (1/2)*Integrate[(1 - Cos[z]^2)*(2 Cos[z])^(2 n), {z, 0, 2 Pi}]/Pi;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 21}] (* Jean-François Alcover, Feb 05 2023 *)

In the MathOverflow link, Nathaniel Johnston conjectures a(n) = A000108(n)*(A000108(n)+1)/2. - Robert Israel, Jan 17 2020

A247306 Array T(n,k) of the expected value of trace(O)^(2k), where O is an n X n orthogonal matrix randomly selected according to Haar measure, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 10, 1, 1, 3, 15, 35, 1, 1, 3, 15, 91, 126, 1, 1, 3, 15, 105, 603, 462, 1, 1, 3, 15, 105, 903, 4213, 1716, 1, 1, 3, 15, 105, 945, 8778, 30537, 6435, 1, 1, 3, 15, 105, 945, 10263, 92235, 227475, 24310, 1

Offset: 1

Nathaniel Johnston, Sep 11 2014

The expected value of trace(O)^k, when k is an odd natural number, is zero.
The first 5 rows of this array are A000012, A001700, A099251, A246860, and A247304.
T(n,k) = A001147(k) when k <= n. When k > n, T(n,k) < A001147(k).

			The upper-left corner of the array begins:
1, 1,  1,   1,   1,     1,      1, ...
1, 3, 10,  35, 126,   462,   1716, ...
1, 3, 15,  91, 603,  4213,  30537, ...
1, 3, 15, 105, 903,  8778,  92235, ...
1, 3, 15, 105, 945, 10263, 127699, ...

L. Pastur and V. Vasilchuk, On the moments of traces of matrices of classical groups, Commun. Math. Phys. 252:149-166, 2004.
MathOverflow, Moments of the trace of orthogonal matrices.

Cf. A001147, A001700, A099251, A246860, and A247304.

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A099251 Bisection of Motzkin sums (A005043).

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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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A246860 Expected value of trace(O)^(2n), where O is a 4 X 4 orthogonal matrix randomly selected according to Haar measure.

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A247306 Array T(n,k) of the expected value of trace(O)^(2k), where O is an n X n orthogonal matrix randomly selected according to Haar measure, read by antidiagonals.

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