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A187306
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Alternating sum of Motzkin numbers A001006.
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9
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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
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OFFSET
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0,3
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COMMENTS
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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
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)
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LINKS
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FORMULA
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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) = (-1)^n*(1-hypergeom([1/2,-n-1],[2],4)). - 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
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MAPLE
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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
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MATHEMATICA
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CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(2x^2(1+x)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 14 2011 *)
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PROG
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(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)
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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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