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%I #78 Feb 23 2023 07:36:24
%S 1,1,3,15,91,603,4213,30537,227475,1730787,13393689,105089229,
%T 834086421,6684761125,54022715451,439742222071,3602118427251,
%U 29671013856627,245613376802185,2042162142208813,17047255430494497,142816973618414817
%N Bisection of Motzkin sums (A005043).
%C 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
%C 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
%C From _Petros Hadjicostas_, Jul 23 2020: (Start)
%C 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_.
%C Eric Weisstein gives some details on how the material in Smith (1968) about isotropic tensors is related to Motzkin sums. (End)
%D G. F. Smith, On isotropic tensors and rotation tensors of dimension m and order n, Tensor (N.S.), Vol. 19 (1968), 79-88 (MR0224008).
%H Vincenzo Librandi, <a href="/A099251/b099251.txt">Table of n, a(n) for n = 0..200</a>
%H D. L. Andrews, <a href="/A005043/a005043.pdf">Letter to N. J. A. Sloane</a>, Apr 10 1978.
%H Georgia Benkart and A. Elduque, <a href="https://arxiv.org/abs/1606.07588">Cross products, invariants, and centralizers</a>, arXiv:1606.07588 [math.RT], 2016.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H MathOverflow, <a href="http://mathoverflow.net/questions/180110/moments-of-the-trace-of-orthogonal-matrices">Moments of the trace of orthogonal matrices</a>.
%H G. F. Smith, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-251fc517-c606-48d2-ac61-d3f22eef4e6c/c/bcp15_1_17.pdf">Lectures on constitutive expressions</a>, Mathematical models and methods in mechanics, pp. 645-678, Banach Center Publ., 15, PWN, Warsaw, 1985 (MR0874855). See p. 653.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IsotropicTensor.html">Isotropic tensor</a>.
%F 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
%F a(n) ~ 3^(2*n + 3/2)/(16*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F 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
%F 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
%F a(n) = hypergeom([1/2, -2*n], [2], 4). - _Peter Luschny_, Jul 25 2020
%p G := (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)): Gser := series(G,x=0,60):
%p 1, seq(coeff(Gser, x^(2*n)), n=1..25); # _Emeric Deutsch_
%p a := n -> hypergeom([1/2, -2*n], [2], 4):
%p seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Jul 25 2020
%t 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 *)
%o (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
%o (Maxima)
%o 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*/
%Y Cf. A005043, A099252, A246860, A247304.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Nov 16 2004
%E More terms from _Emeric Deutsch_, Nov 18 2004
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