A071721 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.

1, 2, 6, 18, 56, 180, 594, 2002, 6864, 23868, 83980, 298452, 1069776, 3863080, 14040810, 51325650, 188574240, 695987820, 2579248980, 9593714460, 35804293200, 134032593240, 503154100020, 1893689067348, 7144084508256

Offset

0,2

Comments

a(n) = A138156(n) - 4*A138156(n-1). - Alzhekeyev Ascar M, Jul 19 2011

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

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

From Colin Defant, Sep 15 2018: (Start)

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.

a(n) is the number of permutations on [n+1] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421. (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.

Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.

Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 32.

Boram Park and Seonjeong Park, Shellable posets arising from even subgraphs of a graph, arXiv preprint arXiv:1705.06423 [math.CO], 2017.

Seonjeong Park, Real toric manifolds and shellable posets arising from graphs, 2018.

Formula

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

a(n) = n*(n+1)*hypergeom([1-n, 2-n], [4], 1) for n>=1. - Peter Luschny, Nov 19 2014

-(n+2)*(n-1)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 18 2017

a(n) = 2*Cat(n+1) - 2*Cat(n) = 2*A000245(n) for n>=1. - Colin Defant, Jun 27 2018

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 23/18 + 7*Pi/(27*sqrt(3)).

Sum_{n>=0} (-1)^n/a(n) = 43/50 - 82*sqrt(5)*log(phi)/375, where phi is the golden ratio (A001622). (End)

From Michael Somos, Apr 22 2022: (Start)

G.f.: (1 - 3*x + x^2 - (1 - x) * sqrt(1 - 4*x))/x^2.

G.f.: (2 - 2*x + x^2)/(1 - 3*x + x^2 + (1 - x)*sqrt(1 - 4*x)).

G.f.: 1 + 1/((1 - x)/(1 - sqrt(1 - 4*x)) - 1/2).

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.

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)

Example

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

Maple

a := n -> `if`(n=0, 1, 6*binomial(2*n, n-1)/(n+2));
seq(a(n), n=0..24); # Peter Luschny, Jun 28 2018

Mathematica

Join[{1}, Table[6n CatalanNumber[n]/(n+2), {n, 30}]] (* Harvey P. Dale, Jun 05 2012 *)
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 *)

Prog

(Sage)
a = lambda n: n*(n+1)*hypergeometric([1-n, 2-n], [4], 1) if n>0 else 1
[simplify(a(n)) for n in range(25)] # Peter Luschny, Nov 19 2014
(PARI) {a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* Michael Somos, Apr 22 2022 */

Crossrefs

Cf. A000108, A000245, A001622, A002421, A068875.

Row sums of triangles A319251, A319252.

Sequence in context: A275857 A111961 A190861 * A125306 A352076 A209797

Adjacent sequences: A071718 A071719 A071720 * A071722 A071723 A071724

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved