A071722 -id:A071722 - cp's OEIS frontend

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) 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

N. J. A. Sloane, Jun 06 2002

nonn

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)

			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

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.

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).

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

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 *)

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

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

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)

A071723 Expansion of (1+x^2C^2)C^4, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 4, 15, 54, 192, 682, 2431, 8710, 31382, 113696, 414086, 1515516, 5571750, 20569590, 76228095, 283481670, 1057628550, 3957577800, 14849601090, 55859886420, 210622646520, 795898303668, 3013646759910, 11432740177564, 43448822603452, 165396657221152

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), A071722 (m=3), this sequence (m=4).

Cf. A000108.

a := n -> (2*(2*n + 1)*(11*n^2 + 17*n + 12)*binomial(2*n, n))/((n + 1)*(n + 2)*(n + 3)*(n + 4)): seq(a(n), n = 0..25); # Peter Luschny, Dec 01 2024

a(n):=sum((k+1)*(k^2+k+1)*binomial(2*n-k,n),k,0,n)/(n+1); /* Vladimir Kruchinin, Sep 28 2011 */

a(n):=(4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4); /* Tani Akinari, Dec 01 2024 */

a(n) = (Sum_{k=0..n} (k+1)*(k^2+k+1)*binomial(2*n-k,n))/(n+1). - Vladimir Kruchinin, Sep 28 2011
a(n) = (4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4). - Tani Akinari, Dec 01 2024
D-finite with recurrence 2*(n+4)*a(n) +2*(-7*n-16)*a(n-1) +3*(9*n+4)*a(n-2) +6*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jul 13 2025

A228178 The number of boundary edges for all ordered trees with n edges.

Original entry on oeis.org

1, 4, 14, 47, 157, 529, 1805, 6238, 21812, 77062, 274738, 987276, 3572568, 13007398, 47617798, 175171543, 647227453, 2400843823, 8937670603, 33380986153, 125045165773, 469700405533, 1768752809221, 6676088636479, 25252913322299, 95712549267151, 363441602176007, 1382467779393307, 5267219868722803

Offset: 0

Louis Shapiro, Aug 20 2013

nonn

Apparently partial sums of A071722. - R. J. Mathar, Aug 25 2013

			The  5 ordered trees with 3 edges have 3,3,2,3,3 boundary edges with UDUDUD having but 2.

Cf. A000108.

x = 'x + O('x^66);
C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
gf = (x*C+2*x^2*C^4)/(1-x);
Vec(gf) \\ Joerg Arndt, Aug 21 2013

G.f.: (x*C+2*x^2*C^4)/(1-x) where C is the g.f. for the Catalan numbers A000108.

Conjecture: 2*(n+3)*a(n) +2*(-7*n-11)*a(n-1) +(29*n+7)*a(n-2) +(-21*n+19)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Aug 25 2013

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A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

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A071723 Expansion of (1+x^2C^2)C^4, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

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A228178 The number of boundary edges for all ordered trees with n edges.

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cp's OEIS Frontend

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.

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A071723 Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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Maxima

Maxima

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A228178 The number of boundary edges for all ordered trees with n edges.

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Author

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Comments

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PARI

Formula

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

A071723 Expansion of (1+x^2C^2)C^4, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.