This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A187306 #81 Nov 16 2022 08:53:55
%S A187306 1,0,2,2,7,14,37,90,233,602,1586,4212,11299,30536,83098,227474,625993,
%T A187306 1730786,4805596,13393688,37458331,105089228,295673995,834086420,
%U A187306 2358641377,6684761124,18985057352,54022715450,154000562759,439742222070,1257643249141
%N A187306 Alternating sum of Motzkin numbers A001006.
%C A187306 Diagonal sums of A089942.
%C A187306 Hankel transform is A187307.
%C A187306 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
%C A187306 Convolution of A005043 with itself. - _Philippe Deléham_, Jan 28 2014
%C A187306 From _Gus Wiseman_, Nov 15 2022: (Start)
%C A187306 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:
%C A187306 (o) . (ooo) (oooo) (ooooo) (oooooo)
%C A187306 ((oo)) ((ooo)) ((oo)oo) ((oo)ooo)
%C A187306 ((oooo)) ((ooo)oo)
%C A187306 (o(oo)o) ((ooooo))
%C A187306 (oo(oo)) (o(oo)oo)
%C A187306 (((oo)o)) (o(ooo)o)
%C A187306 ((o(oo))) (oo(oo)o)
%C A187306 (oo(ooo))
%C A187306 (ooo(oo))
%C A187306 (((oo)oo))
%C A187306 (((ooo)o))
%C A187306 ((o(oo)o))
%C A187306 ((o(ooo)))
%C A187306 ((oo(oo)))
%C A187306 (End)
%H A187306 Vincenzo Librandi, <a href="/A187306/b187306.txt">Table of n, a(n) for n = 0..300</a>
%H A187306 M. H. Albert and V. Vatter, <a href="http://arxiv.org/abs/1301.3122">Generating and enumerating 321-avoiding and skew-merged simple permutations</a>, arXiv preprint arXiv:1301.3122 [math.CO], 2013. - _N. J. A. Sloane_, Feb 11 2013
%H A187306 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A187306 Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%F A187306 G.f.: (1-x-sqrt(1-2*x-3*x^2))/(2*x^2*(1+x)).
%F A187306 a(n) = sum(k=0..n, A001006(k)*(-1)^(n-k)).
%F A187306 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
%F A187306 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
%F A187306 a(n) ~ 3^(n+5/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 15 2013
%F A187306 a(n) = (-1)^n*(1-hypergeom([1/2,-n-1],[2],4)). - _Peter Luschny_, Sep 25 2014
%F A187306 a(n) = A005043(n+1) + (-1)^n. - _Peter Luschny_, Sep 25 2014
%F A187306 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
%p A187306 a := n -> (-1)^n*(1-hypergeom([1/2,-n-1],[2],4));
%p A187306 seq(round(evalf(a(n),99)),n=0..30); # _Peter Luschny_, Sep 25 2014
%t A187306 CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(2x^2(1+x)), {x,0,30}], x] (* _Harvey P. Dale_, Jun 14 2011 *)
%o A187306 (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 */
%o A187306 (PARI) Vec(serreverse(x*(1-x)/(1-x+x^2) + O(x^30))^2) \\ _Andrew Howroyd_, Apr 28 2018
%o A187306 (Sage)
%o A187306 def A187306():
%o A187306 a, b, n = 1, 0, 1
%o A187306 yield a
%o A187306 while True:
%o A187306 n += 1
%o A187306 a, b = b, (2*b+3*a)*(n-1)/(n+1)
%o A187306 yield b - (-1)^n
%o A187306 A187306_list = A187306()
%o A187306 [next(A187306_list) for i in range(20)] # _Peter Luschny_, Sep 25 2014
%Y A187306 Cf. A001006, A005043, A089942.
%K A187306 nonn,easy
%O A187306 0,3
%A A187306 _Paul Barry_, Mar 08 2011