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 A129180 #30 Jan 27 2025 02:28:54
%S A129180 0,1,11,85,583,3785,23843,147437,900559,5453457,32816315,196531781,
%T A129180 1172634391,6976059865,41401814099,245230349021,1450162049695,
%U A129180 8563622372129,50510963880299,297627067200821,1752169739791591,10307304302433513,60592569330907523
%N A129180 Total area below all Schroeder paths of semilength n.
%C A129180 A Schroeder path of semilength n 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.
%H A129180 Alois P. Heinz, <a href="/A129180/b129180.txt">Table of n, a(n) for n = 0..1000</a>
%H A129180 Lara Bossinger and Martina Lanini, <a href="https://arxiv.org/abs/1802.04320">Following Schubert varieties under Feigin's degeneration of the flag variety</a>, arXiv:1802.04320 [math.RT], 2018.
%F A129180 a(n) = Sum_{k=0..n^2} k * A129179(n,k).
%F A129180 G.f.: (1+z)*(1-z-sqrt(1-6*z+z^2))^2/(4*z*(1-6*z+z^2)) (obtained by computing (dG/dt)_{t=1} where G=G(t,z) is defined by G(t,z) = 1+z*G(t,z)+t*z*G(t,t^2*z)G(t,z); see A129179).
%F A129180 a(n) = Sum_{k=0..n} 2*A002315(k)*(Sum_{i=0..n-k+1} binomial(n+1-k,i+2)*binomial(n-k+i,i))/(n-k+1). - _Vladimir Kruchinin_, Mar 02 2016
%F A129180 a(n) ~ 1/2 * (1+sqrt(2))^(2*n+1). - _Vaclav Kotesovec_, Mar 03 2016
%F A129180 D-finite with recurrence -(n+1)*(2*n-5)*a(n) +3*(4*n+1)*(2*n-5)*a(n-1) +(-76*n^2+228*n-89)*a(n-2) +3*(2*n-1)*(4*n-13)*a(n-3) -(2*n-1)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A129180 a(2) = 11 because the areas below the Schroeder paths HH, HUD, UDH, UDUD, UHD and UUDD are 0,1,1,2,3 and 4, respectively.
%p A129180 g:=(1+z)*(1-z-sqrt(1-6*z+z^2))^2/4/z/(1-6*z+z^2): gser:=series(g,z=0,30): seq(coeff(gser,z,n),n=0..24);
%t A129180 CoefficientList[Series[(1 + x)*(1 - x - Sqrt[1 - 6*x + x^2])^2/(4*x*(1 - 6*x + x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 03 2016 *)
%o A129180 (Maxima)
%o A129180 a(n):=sum((((sqrt(2)+1)^(2*k+1)-(1-sqrt(2))^(2*k)*sqrt(2)+(1-sqrt(2))^(2*k))*sum(binomial(n+1-k,i+2)*binomial(n-k+i,i),i,0,n-k+1))/(n-k+1),k,0,n); /* _Vladimir Kruchinin_, Mar 02 2016 */
%Y A129180 Cf. A002315, A006319, A129179.
%K A129180 nonn
%O A129180 0,3
%A A129180 _Emeric Deutsch_, Apr 08 2007