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 A191312 #20 Oct 20 2024 02:31:48
%S A191312 1,1,0,1,0,1,1,0,1,1,1,0,2,1,2,1,0,3,2,2,2,1,0,6,3,4,2,4,1,0,10,6,6,4,
%T A191312 4,4,1,0,20,10,12,6,8,4,9,1,0,35,20,20,12,12,8,9,9,1,0,70,35,40,20,24,
%U A191312 12,18,9,23,1,0,126,70,70,40,40,24,27,18,23,23,1,0,252,126,140,70,80,40,54,27,46,23,65
%N A191312 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having abscissa of the first return to the horizontal axis equal to k (assumed to be 0 if there are no such returns).
%C A191312 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191312 Sum_{k>=0} k*T(n,k) = A093387(n+1).
%F A191312 T(n,0)=1; T(n,1)=0;
%F A191312 T(n,k) = binomial(n-k, floor((n-k)/2))*Sum_{j=0..floor(k/2)-1} c(j), where 2<=k<=n and c(j) = binomial(2*j,j)/(j+1) are the Catalan numbers.
%F A191312 G.f.: G(t,z) = 1/(1-z)+(1-sqrt(1-4*t^2*z^2))/((1-t*z)*(1-2*z+sqrt(1-4*z^2))).
%F A191312 For k>=1, g.f. of column 2k is b_{k-1}*z^{2k}*g and of column 2k+1 is b_{k-1}*z^{2*k+1}*g, where g = 2/(1-2*z+sqrt(1-4*z^2)) and b(k) = Sum_{j=0..k-1} c(j) with c(j) = binomial(2*j,j)/(j+1) = A000108(j) (the Catalan numbers).
%e A191312 T(5,3)=2 because we have HUDHH and HUDUD, where U=(1,1), D=(1,-1), H=(1,0).
%e A191312 Triangle starts:
%e A191312 1;
%e A191312 1, 0;
%e A191312 1, 0, 1;
%e A191312 1, 0, 1, 1;
%e A191312 1, 0, 2, 1, 2;
%e A191312 1, 0, 3, 2, 2, 2;
%e A191312 1, 0, 6, 3, 4, 2, 4;
%p A191312 c := proc (j) options operator, arrow: binomial(2*j, j)/(j+1) end proc: T := proc (n, k) if n < k then 0 elif k = 0 then 1 elif k = 1 then 0 else binomial(n-k, floor((1/2)*n-(1/2)*k))*(sum(c(j), j = 0 .. floor((1/2)*k)-1)) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
%p A191312 G := (1-t*z+t^2*z^2*g*C-t^2*z^3*g*C)/((1-z)*(1-t*z)): g := 2/(1-2*z+sqrt(1-4*z^2)): C := ((1-sqrt(1-4*t^2*z^2))*1/2)/(t^2*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%t A191312 c[j_] := Binomial[2j, j]/(j+1);
%t A191312 T[n_, k_] := Which[n < k, 0, k == 0, 1, k == 1, 0, True, Binomial[n-k, Floor[(n-k)/2]]*(Sum[c[j], {j, 0, Floor[k/2]-1}])];
%t A191312 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 22 2024, after first Maple program *)
%Y A191312 Cf. A001405, A191313.
%K A191312 nonn,tabl
%O A191312 0,13
%A A191312 _Emeric Deutsch_, May 30 2011