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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

REFERENCES

G. F. Smith, Title?, Tensor, Vol. 19 (1968), p. 79.

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, A. Elduque, Cross products, invariants, and centralizers, arXiv preprint arXiv:1606.07588 [math.RT], 2016.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

MathOverflow, Moments of the trace of orthogonal matrices

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..2n+1} (C(2*j,j)*(-1)^(j)*C(2*n+1,j+1))/(2*n+1). - Vladimir Kruchinin, Apr 02 2017

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

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 October 17 14:25 EDT 2018. Contains 316281 sequences. (Running on oeis4.)