The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A099251 Bisection of Motzkin sums (A005043). 7
 1, 1, 3, 15, 91, 603, 4213, 30537, 227475, 1730787, 13393689, 105089229, 834086421, 6684761125, 54022715451, 439742222071, 3602118427251, 29671013856627, 245613376802185, 2042162142208813, 17047255430494497, 142816973618414817 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. 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], , 4). - Peter Luschny, Jul 25 2020 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], , 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 *) PROG (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 (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*/ CROSSREFS Cf. A005043, A099252, A246860, A247304. Sequence in context: A034954 A077783 A047019 * A171790 A006632 A159928 Adjacent sequences:  A099248 A099249 A099250 * A099252 A099253 A099254 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Nov 16 2004 EXTENSIONS More terms from Emeric Deutsch, Nov 18 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 10 10:39 EDT 2021. Contains 342845 sequences. (Running on oeis4.)