A110110 - OEIS

%I #49 Jul 26 2022 14:17:01

%S 1,2,4,8,18,38,88,192,450,1002,2364,5336,12642,28814,68464,157184,

%T 374274,864146,2060980,4780008,11414898,26572086,63521352,148321344,

%U 354870594,830764794,1989102444,4666890936,11180805570,26283115038

%N Number of symmetric Schroeder paths of length 2n (A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis).

%C a(n) = A026003(n-1)+A026003(n) (n>=1; indeed, every symmetric Schroeder path of length 2n is either a left factor L of length n-1 of a Schroeder path, followed by a H=(2,0) step and followed by the mirror image of L, or it is a left factor of length n of a Schroeder paths, followed by its mirror image).

%C a(n) is the number of sequences (f(n)) composed of n letters using letters a, b, c with the following rules. Each new sequence is built by adding a letter to the right end of a previous generation sequence. Letters a and b may not be adjacent. The number of c's <= n/2 in each sequence. Example: f(1) = {[a] [b]}, f(2) = {[aa], [ac], [bb], [bc]}, f(3) = {[aaa] [aac] [aca] [acb] [bbb] [bbc] [bcb] [bca]}. - _Roger Ford_, Jul 13 2014

%H G. C. Greubel, <a href="/A110110/b110110.txt">Table of n, a(n) for n = 0..1000</a>

%H Frédéric Bihan, Francisco Santos, Pierre-Jean Spaenlehauer, <a href="https://arxiv.org/abs/1804.5683">A Polyhedral Method for Sparse Systems with many Positive Solutions</a>, arXiv:1804.05683 [math.CO], 2018.

%H S. Samieinia, <a href="http://dx.doi.org/10.4171/PM/1858">The number of continuous curves in digital geometry</a>, Port. Math. 67 (1) (2010) 75-89.

%H Jacob A. Siehler, <a href="http://arxiv.org/abs/1409.3869">Selections Without Adjacency on a Rectangular Grid</a>, arXiv:1409.3869 [math.CO], 2014, p.9.

%F G.f.: (1 + x) * ( -1 + sqrt( 1 - 6*x^2 + x^4) / (1 - 2*x - x^2)) / (2*x). - _Michael Somos_, Feb 07 2011

%F G.f.: (1+z)R(z^2)/[1-zR(z^2)], where R=1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers.

%F a(2*n) = A050146(n+1). - _Michael Somos_, Feb 07 2011

%F From _Roger Ford_, May 25 2014: (Start)

%F a(2*n) = 3*a(2*n-1) - 2*A026003(2*n-2), n>0.

%F a(2*n+1) = a(2*n) + 2*A026003(2*n) - A006318(n).

%F (End)

%F a(n) ~ sqrt(6*sqrt(2)-8) * (1+sqrt(2))^(n+2)/ (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 09 2016

%F D-finite with recurrence (n+1)*a(n) +(n-3)*a(n-1) +2*(-3*n+2)*a(n-2) +2*(-3*n+8)*a(n-3) +(n-5)*a(n-4) +(n-5)*a(n-5)=0. - _R. J. Mathar_, Jul 26 2022

%e a(2)=4 because we have HH, UDUD, UHD and UUDD.

%e 1 + 2*x + 4*x^2 + 8*x^3 + 18*x^4 + 38*x^5 + 88*x^6 + 192*x^7 + 450*x^8 + ...

%p RR:=(1-z^2-sqrt(1-6*z^2+z^4))/2/z^2: G:=(1+z)*RR/(1-z*RR): Gser:=series(G,z=0,36): 1,seq(coeff(Gser,z^n),n=1..33);

%t CoefficientList[Series[(1+x) * (-1 + Sqrt[1 - 6*x^2 + x^4] / (1 - 2*x - x^2)) / (2*x), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Mar 09 2016 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x) * ( -1 + sqrt( 1 - 6*x^2 + x^4 + x^2 * O(x^n)) / (1 - 2*x - x^2)) / (2*x), n))} /* _Michael Somos_, Feb 07 2011 */

%Y Cf. A026003, A006318.

%Y Partial sums of A247630.

%K nonn

%O 0,2

%A _Emeric Deutsch_, Jul 12 2005