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 A114692 #5 Mar 30 2012 17:36:07 %S A114692 1,1,1,2,1,10,1,40,4,1,160,36,1,674,220,8,1,2994,1180,104,1,13872, %T A114692 6056,848,16,1,66336,30760,5680,272,1,324898,156632,34528,2768,32,1, %U A114692 1621178,803096,199552,22224,672,1,8210904,4150444,1122736,156528,8192,64,1 %N A114692 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k returns to the x-axis (0<=k<=floor(n/2)). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. %C A114692 Row n contains 1+floor(n/2) terms. Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..floor(n/2))=A114693(n-2). %F A114692 G.f.=G=1/(1-z+tz-tzR), where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318). %e A114692 T(4,2)=4 because we have UUD(D)UUD(D),UUD(D)UH(D),UH(D)UUD(D) and UH(D)UH(D), where U=(1,1), D=(1,-1) and H=(2,0) (the returns to the x-axis are shown between parentheses). %e A114692 Triangle starts: %e A114692 1; %e A114692 1; %e A114692 1,2; %e A114692 1,10; %e A114692 1,40,4; %e A114692 1,160,36; %p A114692 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z+t*z-t*z*R): Gser:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form %Y A114692 Cf. A001003, A114693. %K A114692 nonn,tabf %O A114692 0,4 %A A114692 _Emeric Deutsch_, Dec 26 2005