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 A132279 #9 Jul 21 2017 13:35:45 %S A132279 1,1,3,6,15,1,36,4,91,17,1,232,60,5,603,206,26,1,1585,676,110,6,4213, %T A132279 2174,444,37,1,11298,6868,1687,182,7,30537,21446,6196,841,50,1,83097, %U A132279 66356,22100,3612,280,8,227475,203914,77138,14833,1455,65,1 %N A132279 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k doublerises (i.e., UU's) (0 <= k <= floor(n/2) - 1 for n >= 2). %C A132279 Row n contains floor(n/2) terms (n>=2). Row sums yield A118720. T(n,0) = A005043(n+2) (the Riordan numbers). %F A132279 G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*G + z^2*(t(G-1-zG-z^2*G) + 1 + zG + z^2*G)G (see explicit expression at the Maple program). %F A132279 G.f.: G = 2/(1-z-2*z^2+t*z^2+sqrt(1-2*z-3*z^2-2*t*z^2+2*t*z^3+t^2*z^4)). - _Olivier Gérard_, Sep 27 2007 %e A132279 Triangle starts: %e A132279 1; %e A132279 1; %e A132279 3; %e A132279 6; %e A132279 15, 1; %e A132279 36, 4; %e A132279 91, 17, 1; %e A132279 232, 60, 5; %e A132279 T(5,1)=4 because we have UUhDD, UUDhD, hUUDD and UUDDh. %p A132279 G:=((1-z-2*z^2+z^2*t-sqrt((1+z-z^2*t)*(1-3*z-z^2*t)))*1/2)/(z^2*(t+z+z^2-z*t-z^2*t)): Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) end do: 1; 1; for n from 2 to 14 do seq(coeff(P[n],t,j),j= 0..floor((1/2)*n)-1) end do; # yields sequence in triangular form %Y A132279 Cf. A005043, A118720. %K A132279 nonn,tabf %O A132279 0,3 %A A132279 _Emeric Deutsch_, Sep 03 2007