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 A129173 #26 Jan 10 2022 10:24:17
%S A129173 0,1,9,58,336,1853,9945,52487,273939,1418567,7303791,37441560,
%T A129173 191287254,974642943,4955123955,25146686730,127424717400,644873878895,
%U A129173 3260055588615,16465301636090,83092583965020,419031686115875
%N A129173 Total area below all skew Dyck paths of semilength n.
%C A129173 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
%H A129173 G. C. Greubel and Vincenzo Librandi, <a href="/A129173/b129173.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)
%H A129173 E. Deutsch, E. Munarini and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%H A129173 Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, <a href="https://doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.
%F A129173 a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).
%F A129173 a(n) - 5*a(n-1) = A002212(n) + A002212(n-1).
%F A129173 G.f.: (1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)).
%F A129173 (n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.
%F A129173 a(n) ~ 6*5^(n-1)*(1-sqrt(5)/sqrt(Pi*n)) . - _Vaclav Kotesovec_, Oct 20 2012
%e A129173 a(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.
%p A129173 a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n],n=0..25);
%t A129173 CoefficientList[Series[(1+x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*(1-5*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A129173 (PARI) z='z +O('z^25); concat([0], Vec((1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)))) \\ _G. C. Greubel_, Feb 10 2017
%Y A129173 Cf. A002212, A129172.
%K A129173 nonn
%O A129173 0,3
%A A129173 _Emeric Deutsch_, Apr 09 2007