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 A114507 #21 Jan 24 2025 18:28:59
%S A114507 1,1,2,4,10,27,79,240,750,2387,7711,25214,83315,277799,933596,3159187,
%T A114507 10755190,36811479,126594819,437220744,1515844359,5273760446,
%U A114507 18406122609,64426136558,226108087891,795486834627,2804993559426,9911529800630,35090946422404,124462137097349
%N A114507 Number of Dyck paths of semilength n having no ascents of length 3.
%C A114507 Also number of ordered trees with n edges that have no vertices of outdegree 3.
%H A114507 Andrew Howroyd, <a href="/A114507/b114507.txt">Table of n, a(n) for n = 0..1000</a>
%F A114507 G.f. G satisfies z^4*G^4-z^3*G^3+zG^2-G+1=0.
%F A114507 a(n-1) = 1/n*sum(j=ceiling((n+1)/2)..n, binomial(n,j)*binomial(4*j-2*n-2,j-1)*(-1)^(n-j)) n>0. - _Vladimir Kruchinin_, Mar 07 2011
%F A114507 D-finite with recurrence 2*n*(26405927*n-73197215)*(2*n+3)*(n+1)*a(n) +2*n*(2*n+1)*(26405927*n^2-273126414*n+480676927)*a(n-1) +4*(-793701648*n^4+4928830819*n^3-11073984912*n^2+10499531162*n-3092762541)*a(n-2) +2*(375778330*n^4-3447814000*n^3+22123257551*n^2-60324066977*n+51211836006)*a(n-3) +2*(12664700570*n^4-145150764350*n^3+621947195977*n^2-1179232268341*n+833841845214)*a(n-4) -3*(n-4)*(11017381441*n^3-111829680906*n^2+390445674963*n-461862831838)*a(n-5) -(n-4)*(n-5)*(30888861033*n^2-148676625095*n+156786419682)*a(n-6) +3206*(n-5)*(n-6)*(18970222*n-55906401)*(n-4)*a(n-7)=0. - _R. J. Mathar_, Jul 26 2022
%e A114507 a(3) = 4 because we have UDUDUD, UDUUDD, UUDDUD and UUDUDD, where U=(1,1), D=(1,-1).
%p A114507 Order:=36: Y:=solve(series((Y-Y^2)/(1-Y^3+Y^4),Y)=z,Y): seq(coeff(Y,z^n),n=1..32); #(Y=zG)
%o A114507 (Maxima) a114507(n):= 1/n*sum(binomial(n,j)*binomial(4*j-2*n-2, j-1) *(-1)^(n-j),j,ceiling((n+1)/2),n); /* Works for n > 0. Returns a(n-1). _Vladimir Kruchinin_, Mar 07 2011 */
%o A114507 (PARI) a(n)={n++; sum(j=n\2+1, n, binomial(n, j)*binomial(4*j-2*n-2, j-1)*(-1)^(n-j))/n} \\ _Andrew Howroyd_, Jan 24 2025
%Y A114507 Cf. A102403, A114506, A114509.
%K A114507 nonn
%O A114507 0,3
%A A114507 _Emeric Deutsch_, Dec 03 2005
%E A114507 a(27) onwards from _Andrew Howroyd_, Jan 24 2025