A238112 Expansion of g.f.: (1-5*x+2*x^2+(2*x-1)*sqrt(x^2-6*x+1))/(4*x).

0, 0, 1, 5, 23, 107, 509, 2473, 12235, 61463, 312761, 1609005, 8355423, 43741635, 230614773, 1223414481, 6525975315, 34981856303, 188341400945, 1018043304661, 5522585343271, 30056208280091, 164066282507501, 898029800045945, 4927810473507803, 27103503645610567, 149393131346947369, 825093297608481533

Offset: 0

Author

N. J. A. Sloane, Mar 04 2014

Keywords

• nonn

Comments

Number of bracketed decomposable averaging words of degree n.

a(n) is the number of dissections of a convex (n+2)-sided polygon by nonintersecting diagonals such that at least one of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Mar 01 2025

Examples

			From _Peter Bala_, Jan 31 2020: (Start)
The sequence may be obtained from the matrix multiplication of A132372 and the sequence of squares A000290:
   / 1           \   / 1 \       /  1 \
  |  1   1        | |  4  |     |   5  |
  |  2   3  1     | |  9  |  =  |  23  |
  |  6  10  5  1  | | 16  |     | 107  |
  | ...           | | ... |     | ...  | (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• Li Guo and Jun Pei, Averaging algebras, Schroeder numbers and rooted trees, arXiv:1401.7386 [math.RA], 2014.

Crossrefs

Cf. A000290, A006318, A132372, A001850.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); [0,0] cat Coefficients(R!( (1-5*x+2*x^2+(2*x-1)*Sqrt(x^2-6*x+1))/(4*x))); // Marius A. Burtea, Feb 02 2020

Maple

CD := n -> LegendreP(n,3): a := n -> ((3*n^2+3*n-6)*CD(n+2) + (34*n^2+52*n+18)* CD(n) + (20-23*n^2-21*n)*CD(n+1))/(4*(n^3-n)):
[0,0,seq(a(n), n=2..27)]; # Peter Luschny, Feb 01 2020

Mathematica

Join[{0, 0},Table[1/4(2GegenbauerC[n,-(1/2),3]-GegenbauerC[1+n,-(1/2),3]),{n, 2, 30}]] (* Benedict W. J. Irwin, Sep 26 2016 *)
CoefficientList[Series[(1-5*x+2*x^2+(2*x-1)*Sqrt[x^2-6*x+1])/(4*x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2016 *)

PARI

my(x='x+O('x^50)); concat([0,0], Vec((1-5*x+2*x^2+(2*x-1)*sqrt(x^2-6*x+1))/(4*x))) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) ~ (1+2*sqrt(2))* sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 05 2014

a(n) = (2*GegenbauerC(n,-1/2,3)-GegenbauerC(n+1,-1/2,3))/4, n>1. - Benedict W. J. Irwin, Sep 26 2016

D-finite with recurrence: (n+1)*a(n) +(-8*n+5)*a(n-1) +(13*n-32)*a(n-2) +2*(-n+4)*a(n-3)=0. - R. J. Mathar, Jan 25 2020

From Peter Bala, Jan 31 2019: (Start)

O.g.f.: A(x) = x^2*(1 + x*S(x))^2/(1 - x*S(x))^3 = x^2*S(x)^2/(1 - x*S(x)), where S(x) = 1 + 2*x + 6*x^2 + 22*x^3 + ... is the o.g.f. for the large Schröder numbers A006318.

Modulo offset differences, the sequence is given by the matrix-by-vector product A132372 * A000290 (regarded as a column vector). See the example below. (End)

a(n) = ((3*n^2+3*n-6)*CD(n+2) + (34*n^2+52*n+18)*CD(n) + (20-23*n^2-21*n)* CD(n+1))/(4*(n^3-n)) where CD(n) are the central Delannoy numbers A001850, for n >= 2. - Peter Luschny, Feb 01 2020

From Peter Bala, Feb 20 2020: (Start)

a(n) = (1/2)*( A006318(n) - 2*A006318(n-1) ) for n >= 1.

O.g.f.: A(x) = (1/2)*( (1 - 2*x)*S(x) - 1 ), where S(x) is the o.g.f. for the large Schröder numbers A006318. (End)

a(n) = A001003(n-1) + 2*Sum_{k=1..n-2} ( A001003(k) * A001003 (n-k-1) ) for n >= 2. - Muhammed Sefa Saydam, Mar 01 2025