A187306 Alternating sum of Motzkin numbers A001006.

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

Offset: 0

Author

Paul Barry, Mar 08 2011

Keywords

• nonn
• easy

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

From Gus Wiseman, Nov 15 2022: (Start)

Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 2 vertices. This would imply a(n) = A284778(n-1) + A005043(n). For example, the a(0) = 1 through a(5) = 14 trees are:

(o) . (ooo) (oooo) (ooooo) (oooooo)

((oo)) ((ooo)) ((oo)oo) ((oo)ooo)

((oooo)) ((ooo)oo)

(o(oo)o) ((ooooo))

(oo(oo)) (o(oo)oo)

(((oo)o)) (o(ooo)o)

((o(oo))) (oo(oo)o)

(oo(ooo))

(ooo(oo))

(((oo)oo))

(((ooo)o))

((o(oo)o))

((o(ooo)))

((oo(oo)))

(End)

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 [math.CO], 2013. - _N. J. A. Sloane_, Feb 11 2013
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.

Crossrefs

Cf. A001006, A005043, A089942.

Programs

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 *)

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 */

PARI

Vec(serreverse(x*(1-x)/(1-x+x^2) + O(x^30))^2) \\ Andrew Howroyd, Apr 28 2018

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

D-finite with recurrence -(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