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

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

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)

Examples

			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
		

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.

Crossrefs

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

Row sums of triangles A319251, A319252.

gf=(1+x^2*C^2)*C^m: A000782 (m=1), this sequence (m=2), A071722 (m=3), A071723 (m=4).

Programs

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 *)

PARI

{a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* Michael Somos, Apr 22 2022 */

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

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

D-finite with recurrence -(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)