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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], [2], 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], [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 *)

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

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