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 A114509 #14 Jan 24 2025 21:45:08 %S A114509 1,1,2,5,13,37,111,345,1104,3611,12016,40548,138414,477076,1657956, %T A114509 5802920,20436910,72369903,257518806,920333307,3302003826,11888979066, %U A114509 42944410207,155576009845,565127618392,2057903975752,7510967300206 %N A114509 Number of Dyck paths of semilength n having no ascents of length 4. %C A114509 Also number of ordered trees with n edges that have no vertices of outdegree 4. %F A114509 G.f.: G=G(z) satisfies z^5*G^5-z^4*G^4+zG^2-G+1=0. %F A114509 a(n) = (1/n)*sum(j=ceiling((3*n+2)/5)..n, C(n,j)*C(5*j-3*n-2,j-1) * (-1)^(n-j)), n>0. [_Vladimir Kruchinin_, Mar 07 2011] %e A114509 a(4) = 13 because among the Catalan(4)=14 Dyck paths of semilength 4 only UUUUDDDD has an ascent of length 4 (here U=(1,1), D=(1,-1)). %p A114509 Order:=35: Y:=solve(series((Y-Y^2)/(1-Y^4+Y^5),Y)=z,Y): seq(coeff(Y,z^n),n=1..30); #(Y=zG) %o A114509 (Maxima) a114509(n):= 1/n*sum(binomial(n,j)*binomial(5*j-3*n-2,j-1)* (-1)^(n-j),j,ceiling((3*n+2)/5),n); /* Works for n > 0. Returns a(n-1). _Vladimir Kruchinin_, Mar 07 2011 */ %Y A114509 Cf. A102403, A114507, A114508. %K A114509 nonn %O A114509 0,3 %A A114509 _Emeric Deutsch_, Dec 03 2005