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

Author

N. J. A. Sloane, Nov 16 2004

Keywords

• nonn
• easy

Comments

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)

References

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

Links

• 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.

Crossrefs

Cf. A005043, A099252, A246860, A247304.

Programs

Maple

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

Mathematica

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 *)

Maxima

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*/

PARI

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

Formula

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

Extensions

More terms from Emeric Deutsch, Nov 18 2004