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%I #76 Dec 19 2023 17:45:07
%S 1,2,6,18,56,180,594,2002,6864,23868,83980,298452,1069776,3863080,
%T 14040810,51325650,188574240,695987820,2579248980,9593714460,
%U 35804293200,134032593240,503154100020,1893689067348,7144084508256
%N Expansion of (1+x^2*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%C a(n) = A138156(n) - 4*A138156(n-1). - _Alzhekeyev Ascar M_, Jul 19 2011
%C Apparently, for n>=1, the sum of the heights of the first and last peaks in all Dyck n-paths (in paths with one peak the height counts as both first and last). - _David Scambler_, Oct 05 2012
%C For n>=1, a(n) is the total number of nonempty subtrees over all binary trees having n+1 internal nodes. Here, a binary tree is a full (each node has two or zero children), rooted, plane (ordered), unlabeled tree. An empty subtree is a tree attached to the root that consists only of an external node. a(n) = 2*A002057(n-2) + A068875(n). - _Geoffrey Critzer_, Sep 16 2013
%C From _Colin Defant_, Sep 15 2018: (Start)
%C a(n) is the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.
%C a(n) is the number of permutations on [n+1] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421. (End)
%H Michael De Vlieger, <a href="/A071721/b071721.txt">Table of n, a(n) for n = 0..1000</a>
%H Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H Stoyan Dimitrov, <a href="https://arxiv.org/abs/2002.12322">On permutation patterns with constrained gap sizes</a>, arXiv:2002.12322 [math.CO], 2020.
%H Ling Gao, <a href="http://hdl.handle.net/20.500.12680/h989rb533">Graph assembly for spider and tadpole graphs</a>, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 32.
%H Boram Park and Seonjeong Park, <a href="https://arxiv.org/abs/1705.06423">Shellable posets arising from even subgraphs of a graph</a>, arXiv preprint arXiv:1705.06423 [math.CO], 2017.
%H Seonjeong Park, <a href="http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2060-06.pdf">Real toric manifolds and shellable posets arising from graphs</a>, 2018.
%F a(n) = 6n * (2n)! / [(n+2)n!(n+1)! ], n>0. In terms of Catalan numbers (A000108), a(n) = 6n*Cat(n)/(n+2), n>0. - _Ralf Stephan_, Mar 11 2004
%F a(n) = n*(n+1)*hypergeom([1-n, 2-n], [4], 1) for n>=1. - _Peter Luschny_, Nov 19 2014
%F -(n+2)*(n-1)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 18 2017
%F a(n) = 2*Cat(n+1) - 2*Cat(n) = 2*A000245(n) for n>=1. - _Colin Defant_, Jun 27 2018
%F From _Amiram Eldar_, Mar 22 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 23/18 + 7*Pi/(27*sqrt(3)).
%F Sum_{n>=0} (-1)^n/a(n) = 43/50 - 82*sqrt(5)*log(phi)/375, where phi is the golden ratio (A001622). (End)
%F From _Michael Somos_, Apr 22 2022: (Start)
%F G.f.: (1 - 3*x + x^2 - (1 - x) * sqrt(1 - 4*x))/x^2.
%F G.f.: (2 - 2*x + x^2)/(1 - 3*x + x^2 + (1 - x)*sqrt(1 - 4*x)).
%F G.f.: 1 + 1/((1 - x)/(1 - sqrt(1 - 4*x)) - 1/2).
%F a(n) = b(n+1) - b(n) for all n in Z if b(0) = 2, b(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A068875.
%F 0 = a(n)*(+16*a(n+1) -58*a(n+2) +18*a(n+3)) +a(n+1)*(+18*a(n+1) +15*a(n+2) -13*a(n+3)) +a(n+2)*(+3*a(n+2) +a(n+3)) for all n in Z if a(0) = 0, a(-1) = 3, a(-2) = -1. (End)
%e G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 180*x^5 + 594*x^6 + 2002*x^7 + ... - _Michael Somos_, Apr 22 2022
%p a := n -> `if`(n=0, 1, 6*binomial(2*n, n-1)/(n+2));
%p seq(a(n), n=0..24); # _Peter Luschny_, Jun 28 2018
%t Join[{1},Table[6n CatalanNumber[n]/(n+2),{n,30}]] (* _Harvey P. Dale_, Jun 05 2012 *)
%t nn=20;t=(1-(1-4x)^(1/2))/(2x);CoefficientList[Series[D[1+x (y t -y+1)^2,y]/.y->1,{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 16 2013 *)
%o (Sage)
%o a = lambda n: n*(n+1)*hypergeometric([1-n, 2-n], [4], 1) if n>0 else 1
%o [simplify(a(n)) for n in range(25)] # _Peter Luschny_, Nov 19 2014
%o (PARI) {a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* _Michael Somos_, Apr 22 2022 */
%Y Cf. A000108, A000245, A001622, A002421, A068875.
%Y Row sums of triangles A319251, A319252.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2002
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