cp's OEIS Frontend

A191528 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have k returns to the axis.

%I A191528 #21 Oct 17 2020 02:17:37
%S A191528 1,1,1,1,2,1,3,2,1,6,3,1,10,6,3,1,20,10,4,1,35,20,10,4,1,70,35,15,5,1,
%T A191528 126,70,35,15,5,1,252,126,56,21,6,1,462,252,126,56,21,6,1,924,462,210,
%U A191528 84,28,7,1,1716,924,462,210,84,28,7,1,3432,1716,792,330,120,36,8,1,6435,3432,1716,792,330,120,36,8,1
%N A191528 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have k returns to the axis.
%C A191528 Row n contains 1+floor(n/2) entries.
%C A191528 Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
%C A191528 T(n,0) = A001405(n-1).
%C A191528 Rows 0, 2, 4, ... form triangle A100100.
%C A191528 Rows 1, 3, 5, ... form triangle A092392.
%C A191528 Sum_{k>=0} k*T(n,k) = A037955(n).
%C A191528 From _Roger Ford_, Oct 16 2020: (Start)
%C A191528 This is an empirical observation. T(n,k) = the number of different semi-meander arch depth models with n+2 top arches and k+1 arches at depth 0.  T(3,1) = the number of different semi-meander arch depth models with 5 top arches and 2 arches at depth 0.
%C A191528 Example: The depth of a semi-meander arch is the number of covering arches directly above the arch.  The arch depth model is the number of arches at each depth starting at 0 for a specific semi-meander.  The following is the arch depth models for semi-meanders with 5 top arches.
%C A191528             /\                                   /\
%C A191528            //\\                                 /  \
%C A191528           ///\\\             depth             //\  \             depth
%C A191528          ////\\\\  /\       (0)(1)(2)(3)      ///\\/\\ /\        (0)(1)(2)
%C A191528 depth    0123      0  model= 2  1  1  1       012  1   0   model= 2  2  1
%C A191528          /\
%C A191528         //\\    /\         depth         /\   /\            depth
%C A191528        ///\\\  //\\       (0)(1)(2)     //\\ //\\ /\       (0)(1)
%C A191528 depth  012     01   model= 2  2  1      01   01   0  model= 3  2
%C A191528          /\
%C A191528         /  \               depth
%C A191528        //\/\\ /\ /\       (0)(1)
%C A191528 depth  01 1   0  0  model= 3  2
%C A191528 There are 5 more semi-meanders with 5 top arches.  They are reflections of the above semi-meanders over a center vertical line and they yield the same arch depth models as the semi-meanders above.
%C A191528 T(3,1) = 2    different models=  2 2 1 and 2 1 1 1;
%C A191528 T(3,2) = 1     model=  3 2    (End).
%H A191528 Indranil Ghosh, <a href="/A191528/b191528.txt">Rows 0..100, flattened</a>
%F A191528 T(n,k) = binomial(n-k-1, ceiling(n/2)-1) if 0 <= k <= floor(n/2).
%F A191528 G.f.:  G(t,z) = 1/((1-z*c)*(1-t*z^2*c)), where c = (1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.
%e A191528 T(6,2)=3 because we have U(D)U(D)UU, U(D)UUD(D), and UUD(D)U(D), where U=(1,1) and D=(1,-1) (the return steps to the axis are shown between parentheses).
%e A191528 Triangle starts:
%e A191528    1:
%e A191528    1;
%e A191528    1, 1;
%e A191528    2, 1;
%e A191528    3, 2, 1;
%e A191528    6, 3, 1;
%e A191528   10, 6, 3, 1;
%p A191528 T := proc (n, k) if k <= floor((1/2)*n) then binomial(n-k-1, ceil((1/2)*n)-1) else 0 end if end proc: for n from 0 to 16 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%t A191528 Flatten[Table[Binomial[n-k-1,Ceiling[(n/2)-1]],{n,0,16},{k,0,Floor[n/2]}]] (* _Indranil Ghosh_, Mar 05 2017 *)
%o A191528 (PARI)
%o A191528 tabf(nn) = if(n==0, print1(1,", "), {for (n=1, nn, for(k=0, floor(n/2), print1(binomial(n-k-1, ceil((n/2)-1)),", ");); print();); });
%o A191528 tabf(16); \\ _Indranil Ghosh_, Mar 05 2017
%Y A191528 Cf. A001405, A037955, A092392, A100100.
%K A191528 nonn,tabf
%O A191528 0,5
%A A191528 _Emeric Deutsch_, Jun 06 2011