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 A129167 #19 Jul 23 2017 12:15:06
%S A129167 0,1,3,9,30,109,420,1685,6960,29391,126291,550359,2426502,10803801,
%T A129167 48507843,219377949,998436792,4569488371,21016589073,97090411019,
%U A129167 450314942682,2096122733211,9788916220518,45850711498859,215348942668680,1013979873542689,4785437476592805,22633143884165985,107258646298581390
%N A129167 Number of base pyramids in all skew Dyck paths of semilength n.
%C A129167 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. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
%C A129167 a(n) = |A091699(n+1)|. Partial sums of A033321(n), n = 1, 2, 3, ....
%C A129167 a(n+1) is the number of 3-colored Motzkin paths of length n with no peaks at level 1. - _José Luis Ramírez Ramírez_, Mar 31 2013
%H A129167 Vincenzo Librandi, <a href="/A129167/b129167.txt">Table of n, a(n) for n = 0..200</a>
%H A129167 E. Deutsch, E. Munarini, 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
%F A129167 a(n) = Sum_{k=0..n} k*A129165(n,k).
%F A129167 G.f.: (1 - 3*z - sqrt(1 - 6*z + 5*z^2))/(z*(3 - 3*z - sqrt(1 - 6*z + 5*z^2))).
%F A129167 Recurrence: 2*(n+1)*a(n) = (13*n-3)*a(n-1) - 4*(4*n-3)*a(n-2) + 5*(n-1)*a(n-3) . - _Vaclav Kotesovec_, Oct 20 2012
%F A129167 a(n) ~ 5^(n+5/2)/(72*sqrt(Pi)*n^(3/2)) . - _Vaclav Kotesovec_, Oct 20 2012
%e A129167 a(2)=3 because in the paths (UD)(UD), (UUDD) and UUDL we have altogether 3 base pyramids (shown between parentheses).
%p A129167 G:=(1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t A129167 CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(x*(3-3*x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A129167 (PARI) z='z+O('z^66); concat([0], Vec((1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)))) \\ _Joerg Arndt_, Aug 27 2014
%Y A129167 Cf. A033321, A091699, A129165.
%K A129167 nonn
%O A129167 0,3
%A A129167 _Emeric Deutsch_, Apr 04 2007