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 A216318 #24 Oct 30 2020 12:20:14
%S A216318 0,1,2,8,31,119,456,1749,6721,25883,99892,386308,1496782,5809478,
%T A216318 22584160,87922215,342741285,1337698515,5226732060,20442936360,
%U A216318 80031775890,313585934610,1229695855440,4825705232010,18950613058026,74467158658974,292797216620776,1151895428382104
%N A216318 Number of peaks in all Dyck n-paths after changing each valley to a peak by the transform DU -> UD.
%H A216318 Vincenzo Librandi, <a href="/A216318/b216318.txt">Table of n, a(n) for n = 0..1000</a>
%F A216318 a(0)=0, a(1)=1, a(n>=2) = A001700(n-1) - Sum_{k=0..n-3} A001700(k) + Sum_{k=0..n-2} A003516(k) - 1.
%F A216318 G.f.: (16*x*(1+sqrt(1-4*x)+(5+3*sqrt(1-4*x)-2*x) * (-1+x)*x)) / ((1+sqrt(1-4*x))^5 * sqrt(1-4*x)).
%F A216318 a(n) ~ 5*2^(2*n-3)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 21 2014
%F A216318 a(n) = C(2*n-2,n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)), n>1, a(0)=0, a(1)=1. - _Vladimir Kruchinin_, Oct 30 2020
%e A216318 The 5 Dyck 3-paths after changing DU to UD become two copies of UUUDDD with one peak each and three copies of UUDUDD with two peaks each giving a(3)=8.
%t A216318 CoefficientList[Series[(16*x*(1+Sqrt[1-4*x]+(5+3*Sqrt[1-4*x]-2*x)*(-1+x) x))/((1+Sqrt[1-4*x])^5*Sqrt[1-4*x]),{x,0,27}],x]
%o A216318 (PARI) x='x+O('x^50); concat([0], Vec((16*x*(1+sqrt(1-4*x)-(5+3*sqrt(1-4*x)-2*x)*(1-x)*x)) / ((1+sqrt(1-4*x))^5*sqrt(1-4*x)))) \\ _G. C. Greubel_, Apr 01 2017
%o A216318 (Maxima)
%o A216318 a(n):=if n<2 then n else binomial(2*n-2,n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)); /* _Vladimir Kruchinin_, Oct 30 2020 */
%Y A216318 Cf. A001700, A003516, A005891.
%K A216318 nonn
%O A216318 0,3
%A A216318 _David Scambler_, Sep 03 2012