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 A121320 #20 Apr 07 2020 03:42:13
%S A121320 0,0,1,2,6,18,59,203,724,2643,9802,36755,138935,528406,2019419,
%T A121320 7748125,29825844,115132729,445498768,1727434607,6710501025,
%U A121320 26110567532,101744332967,396983837719,1550777652546,6064476854065,23739056348161
%N A121320 Number of vertices in all ordered (plane) trees with n edges that are at distance two from all the leaves above them.
%H A121320 Andrew Howroyd, <a href="/A121320/b121320.txt">Table of n, a(n) for n = 0..500</a>
%F A121320 G.f.: x^2*(1 + 1/sqrt(1 - 4*x))/(2 - 2*x - 2*x^2). - Reformulated by _Georg Fischer_, Apr 06 2020
%F A121320 Conjecture: (-n+2)*a(n) +(5*n-12)*a(n-1) +(-3*n+8)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jun 22 2016
%F A121320 a(n) ~ 2^(2*n-1) / (11*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 07 2020
%e A121320 a(4)=6 since the root has the distance two property for the trees uudduudd and uudududd. There are similar points at height 1 for uuududdd, uuudddud and uduuuddd. The distance two point is at height 2 for uuuudddd.
%t A121320 CoefficientList[Series[x^2(1 + 1/Sqrt[1 - 4x])/(2(1 - x - x^2)), {x, 0, 26}], x] (* _Robert G. Wilson v_, Aug 21 2006 *)
%o A121320 (PARI) seq(n)={Vec(x^2*(1 + 1/sqrt(1 - 4*x + O(x^(n-1))))/(2 - 2*x - 2*x^2), -(n+1))} \\ _Andrew Howroyd_, Apr 06 2020
%Y A121320 Cf. A000045, A024718.
%K A121320 easy,nonn
%O A121320 0,4
%A A121320 _Louis Shapiro_, Aug 25 2006
%E A121320 More terms from _Robert G. Wilson v_, Aug 21 2006