OFFSET
0,4
COMMENTS
Number of bracketed decomposable averaging words of degree n.
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.
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)
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)
EXAMPLE
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)
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 *)
PROG
(PARI) 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
(Magma) R<x>:=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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 04 2014
STATUS
approved