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A238112
Expansion of g.f.: (1-5*x+2*x^2+(2*x-1)*sqrt(x^2-6*x+1))/(4*x).
2
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
0,4
COMMENTS
Number of bracketed decomposable averaging words of degree n.
LINKS
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)
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)
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