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 A247287 #13 Jul 24 2022 12:41:21
%S A247287 0,0,1,4,13,38,108,304,857,2426,6902,19728,56622,163092,471205,
%T A247287 1365008,3963321,11530786,33607190,98105616,286795300,839470664,
%U A247287 2460038427,7216652488,21190820678,62279238828,183185851903,539220930004,1588341106957,4681678922366
%N A247287 Number of weak peaks in all Motzkin paths of length n. A weak peak of a Motzkin path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the Motzkin path u*duu*h*h*dd, where u=(1,1), h=(1,0), d(1,-1), has 4 weak peaks (shown by the stars).
%C A247287 a(n) = Sum(k*A247286(n,k), 0<=k<=n).
%H A247287 Alois P. Heinz, <a href="/A247287/b247287.txt">Table of n, a(n) for n = 0..1000</a>
%F A247287 G.f.: (1-z-sqrt(1-2*z-3*z^2))/(2*(1-z)^2*sqrt(1-2*z-3*z^2)).
%F A247287 a(n) ~ 3^(n+3/2) / (8*sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 16 2014
%F A247287 D-finite with recurrence n*a(n) +(-5*n+3)*a(n-1) +2*(3*n-4)*a(n-2) +2*(n-1)*a(n-3) +(-7*n+16)*a(n-4) +3*(n-3)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2022
%e A247287 a(3)=4 because the Motzkin paths hhh, hu*d, u*dh, and u*h*d have 0, 1, 1, and 2 weak peaks (shown by the stars).
%p A247287 g := ((1-z-sqrt(1-2*z-3*z^2))*(1/2))/((1-z)^2*sqrt(1-2*z-3*z^2)): gser := series(g, z = 0, 34): seq(coeff(gser, z, n), n = 0 .. 32);
%o A247287 (PARI) z='z+O('z^66); concat([0,0],Vec((1-z-sqrt(1-2*z-3*z^2))/(2*(1-z)^2*sqrt(1-2*z-3*z^2)))) \\ _Joerg Arndt_, Sep 14 2014
%Y A247287 Cf. A001006, A247286.
%K A247287 nonn
%O A247287 0,4
%A A247287 _Emeric Deutsch_, Sep 14 2014