# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a134423 Showing 1-1 of 1 %I A134423 #3 Mar 30 2012 17:36:15 %S A134423 1,1,2,1,3,2,1,5,5,3,2,1,8,10,8,6,5,2,1,13,20,19,17,16,11,7,3,2,1,21, %T A134423 38,42,42,43,36,29,18,12,8,5,2,1,34,71,89,98,108,102,92,72,55,40,29, %U A134423 20,13,7,3,2,1,55,130,182,218,255,264,258,228,195,158,125,96,74,52,35,22,14,8 %N A134423 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), such that the area between the x-axis and the path is k (n>=0, 0<=k<=floor(n^2/4)). %C A134423 Row n has 1+floor(n^2/4) terms. Row sums yield A128720. T(n,0)=fibonacci(n+1) (A000045). T(n,1)=A001629(n). Sum(k*T(n,k),k>=0)=A134424(n). %F A134423 G.f.=G(t,z) satisfies G(t,z)=1/[1-z-z^2-(tz^2)G(t,tz)]. Rec. rel. for the row generating polynomials P[n]=P[n](t): P[n]=P[n-1]+P[n-2]+Sum(t^(j+1)P[j]P[n-2-j], j=0..n-2) for n>=2; P[0]=P[1]=1. %e A134423 T(4,2)=3 because we have hUhD, UhDh and UDUD. %e A134423 Triangle starts: %e A134423 1; %e A134423 1; %e A134423 2,1; %e A134423 3,2,1; %e A134423 5,5,3,2,1; %e A134423 8,10,8,6,5,2,1; %p A134423 P[0]:=1: P[1]:=1: for n from 2 to 9 do P[n]:=sort(expand(P[n-1]+P[n-2]+sum(P[j]*P[n-2-j]*t^(j+1),j=0..n-2))) end do: for n from 0 to 9 do seq(coeff(P[n], t, j),j=0..floor((1/4)*n^2)) end do; # yields sequence in triangular form %Y A134423 Cf. A128720, A000045, A001629, A134424. %K A134423 nonn,tabf %O A134423 0,3 %A A134423 _Emeric Deutsch_, Oct 25 2007 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE