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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).

%I A238112 #53 Mar 10 2025 16:39:54
%S A238112 0,0,1,5,23,107,509,2473,12235,61463,312761,1609005,8355423,43741635,
%T A238112 230614773,1223414481,6525975315,34981856303,188341400945,
%U A238112 1018043304661,5522585343271,30056208280091,164066282507501,898029800045945,4927810473507803,27103503645610567,149393131346947369,825093297608481533
%N A238112 Expansion of g.f.: (1-5*x+2*x^2+(2*x-1)*sqrt(x^2-6*x+1))/(4*x).
%C A238112 Number of bracketed decomposable averaging words of degree n.
%C A238112 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
%H A238112 G. C. Greubel, <a href="/A238112/b238112.txt">Table of n, a(n) for n = 0..1000</a>
%H A238112 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A238112 Li Guo and Jun Pei, <a href="http://arxiv.org/abs/1401.7386">Averaging algebras, Schroeder numbers and rooted trees</a>, arXiv:1401.7386 [math.RA], 2014.
%F A238112 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
%F A238112 a(n) = (2*GegenbauerC(n,-1/2,3)-GegenbauerC(n+1,-1/2,3))/4, n>1. - _Benedict W. J. Irwin_, Sep 26 2016
%F A238112 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
%F A238112 From _Peter Bala_, Jan 31 2019: (Start)
%F A238112 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.
%F A238112 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)
%F A238112 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
%F A238112 From _Peter Bala_, Feb 20 2020: (Start)
%F A238112 a(n) = (1/2)*( A006318(n) - 2*A006318(n-1) ) for n >= 1.
%F A238112 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)
%F A238112 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
%e A238112 From _Peter Bala_, Jan 31 2020: (Start)
%e A238112 The sequence may be obtained from the matrix multiplication of A132372 and the sequence of squares A000290:
%e A238112    / 1           \   / 1 \       /  1 \
%e A238112   |  1   1        | |  4  |     |   5  |
%e A238112   |  2   3  1     | |  9  |  =  |  23  |
%e A238112   |  6  10  5  1  | | 16  |     | 107  |
%e A238112   | ...           | | ... |     | ...  | (End)
%p A238112 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)):
%p A238112 [0,0,seq(a(n), n=2..27)]; # _Peter Luschny_, Feb 01 2020
%t A238112 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 *)
%t A238112 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 *)
%o A238112 (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
%o A238112 (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
%Y A238112 Cf. A000290, A006318, A132372, A001850.
%K A238112 nonn
%O A238112 0,4
%A A238112 _N. J. A. Sloane_, Mar 04 2014