This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138164 #35 Nov 15 2021 05:32:20
%S A138164 1,1,2,4,9,20,47,109,262,622,1516,3653,8988,21883,54213,133004,331233,
%T A138164 817432,2044151,5068346,12716872,31651555,79636493,198843284,
%U A138164 501466519,1255489165,3172569392,7961388439,20152910577,50674576772
%N A138164 Row sums of Riordan array (c(-x^2),xc(-x^2)^2)^(-1) where c(x) is the g.f. of A000108.
%C A138164 Hankel transform is A005161.
%C A138164 A transform of the F(n+1) by (1,x(1-x^2))^(-1).
%C A138164 (c(-x^2),xc(-x^2)^2)^(-1) factors as (1,x(1-x^2))^(-1)*(1/(1-x^2),x/(1-x^2)).
%C A138164 It appears that a(n) is the number of Dyck paths (A000108) of semilength n in which all non-return descents are of even length (a return descent is a maximal sequence of downsteps that returns the path to ground level). For example, a(4) = 9 counts, among others, UUUDDUDD and UDUUUDDD but not UUDUUDDD. - _David Callan_, Nov 13 2021
%H A138164 Vaclav Kotesovec, <a href="/A138164/b138164.txt">Table of n, a(n) for n = 0..2000</a>
%H A138164 J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, <a href="http://jl.baril.u-bourgogne.fr/cartes.pdf">Rooted planar maps modulo some patternss</a>, Preprint 2016.
%F A138164 G.f.: 1/(1-v-v^2) where v=(2/sqrt(3))*sin(arcsin(x*3*sqrt(3)/2)/3) is the series reversion of x(1-x^2). [Corrected by _Paul D. Hanna_, Feb 24 2015]
%F A138164 From _Gary W. Adamson_, Jul 07 2011: (Start)
%F A138164 Let M = the production matrix:
%F A138164 1, 1, 0, 0, 0, 0, 0, 0, ...
%F A138164 1, 0, 1, 0, 0, 0, 0, 0, ...
%F A138164 1, 1, 0, 1, 0, 0, 0, 0, ...
%F A138164 1, 0, 1, 0, 1, 0, 0, 0, ...
%F A138164 1, 1, 0, 1, 0, 1, 0, 0, ...
%F A138164 1, 0, 1, 0, 1, 0, 1, 0, ...
%F A138164 1, 1, 0, 1, 0, 1, 0, 1, ...
%F A138164 ...
%F A138164 a(n) = top left term of M^n. a(n+1) = sum of top row terms of M^n. Example: top row of M^3 = (4, 3, 1, 1), where a(3) = 4 and a(4) = 9 = (4 + 3 + 1 + 1). (End)
%F A138164 v(x) = Sum_{n>=1} A001764(n-1)*x^(2*n-1). - _Paul D. Hanna_, Feb 24 2015
%F A138164 Conjecture: -8*n*(n-1)*a(n) + 12*(n+3)*(n-1)*a(n-1) + 2*(89*n^2-512*n+651)*a(n-2) + 3*(-127*n^2+715*n-966)*a(n-3) + 3*(-247*n^2+2431*n-5698)*a(n-4) + 9*(75*n-341)*(3*n-16)*a(n-5) - 72*(3*n-14)*(3*n-16)*a(n-6) = 0. - _R. J. Mathar_, Feb 24 2015
%F A138164 From _Vaclav Kotesovec_, Nov 15 2021: (Start)
%F A138164 Recurrence (of order 4): 4*(n-1)*n*(11*n^2 - 50*n + 48)*a(n) = 12*(n-1)*(11*n^3 - 50*n^2 + 66*n - 40)*a(n-1) + (253*n^4 - 2294*n^3 + 6379*n^2 - 5898*n + 720)*a(n-2) - 3*(297*n^4 - 2538*n^3 + 7347*n^2 - 7946*n + 2000)*a(n-3) + 3*(3*n - 10)*(3*n - 8)*(11*n^2 - 28*n + 9)*a(n-4).
%F A138164 a(n) ~ (45*(1 - (-1)^n) + 26*sqrt(3)*(1 + (-1)^n)) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). (End)
%p A138164 vx := 2/sqrt(3)*sin(arcsin(x*3*sqrt(3)/2)/3) ;
%p A138164 A138164 := proc(n)
%p A138164 1/(1-vx-vx^2) ;
%p A138164 coeftayl(%,x=0,n) ;
%p A138164 subs(4^(1/2)=2,%) ;
%p A138164 end proc: # _R. J. Mathar_, Jul 28 2016
%t A138164 CoefficientList[Series[1/(1/3 - 2*Sin[1/3*ArcSin[3*Sqrt[3]*x/2]]/Sqrt[3] + 2*Cos[2/3*ArcSin[3*Sqrt[3]*x/2]]/3), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Nov 15 2021 *)
%o A138164 (PARI) my(x='x+O('x^66)); v=serreverse(x*(1-x^2)); Vec(1/(1-v-v^2)) \\ _Joerg Arndt_, Feb 24 2015
%K A138164 nonn
%O A138164 0,3
%A A138164 _Paul Barry_, Mar 03 2008