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A187306 Alternating sum of Motzkin numbers A001006. 5
1, 0, 2, 2, 7, 14, 37, 90, 233, 602, 1586, 4212, 11299, 30536, 83098, 227474, 625993, 1730786, 4805596, 13393688, 37458331, 105089228, 295673995, 834086420, 2358641377, 6684761124, 18985057352, 54022715450, 154000562759, 439742222070, 1257643249141 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Diagonal sums of A089942.

Hankel transform is A187307.

Also gives the number of simple permutations of each length that avoid the pattern 321 (i.e. are the union of two increasing sequences, and in one line notation contain no nontrivial block of values which form an interval). There are 2 such permutations of length 4, 2 of length 5, etc. - Michael Albert, Jun 20 2012

Convolution of A005043 with itself. - Philippe Deléham, Jan 28 2014

REFERENCES

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

M. H. Albert and V. Vatter, Generating and enumerating 321-avoiding and skew-merged simple permutations, arXiv preprint arXiv:1301.3122, 2013. - N. J. A. Sloane, Feb 11 2013

Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657, 2014

FORMULA

G.f.: (1-x-sqrt(1-2*x-3*x^2))/(2*x^2*(1+x)).

a(n) = sum(k=0..n, A001006(k)*(-1)^(n-k)).

Conjecture: -(n+2)*a(n) +(n-1)*a(n-1) +(5*n-2)*a(n-2) +3*(n-1)a(n-3)=0. - R. J. Mathar, Nov 17 2011

a(n) = (2*sum(j=0..n, C(2*j+1,j+1)*(-1)^(n-j)*C(n+2,j+2)))/(n+2). - Vladimir Kruchinin, Feb 06 2013

a(n) ~ 3^(n+5/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 15 2013

a(n) = (-1)^n*(1-hypergeom([1/2,-n-1],[2],4)). - Peter Luschny, Sep 25 2014

a(n) = A005043(n+1) + (-1)^n. - Peter Luschny, Sep 25 2014

G.f.: (1/(1 - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))))^2, a continued fraction. - Ilya Gutkovskiy, Sep 23 2017

MAPLE

a := n -> (-1)^n*(1-hypergeom([1/2, -n-1], [2], 4));

seq(round(evalf(a(n), 99)), n=0..30); # Peter Luschny, Sep 25 2014

MATHEMATICA

CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(2x^2(1+x)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 14 2011 *)

PROG

(PARI) x='x+O('x^66); Vec((1-x-sqrt(1-2*x-3*x^2))/(2*x^2*(1+x))) /* Joerg Arndt, Mar 07 2013 */

(Sage)

def A187306():

    a, b, n = 1, 0, 1

    yield a

    while True:

        n += 1

        a, b = b, (2*b+3*a)*(n-1)/(n+1)

        yield b - (-1)^n

A187306_list = A187306()

[A187306_list.next() for i in range(20)] # Peter Luschny, Sep 25 2014

CROSSREFS

Cf. A001006, A005043, A089942.

Sequence in context: A156435 A228432 A162460 * A061274 A061575 A290646

Adjacent sequences:  A187303 A187304 A187305 * A187307 A187308 A187309

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 08 2011

STATUS

approved

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Last modified November 23 00:33 EST 2017. Contains 295107 sequences.