OFFSET
0,5
COMMENTS
Row n contains 1+floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
T(n,0) = A001405(n-1).
Rows 0, 2, 4, ... form triangle A100100.
Rows 1, 3, 5, ... form triangle A092392.
Sum_{k>=0} k*T(n,k) = A037955(n).
From Roger Ford, Oct 16 2020: (Start)
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.
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.
/\ /\
//\\ / \
///\\\ depth //\ \ depth
////\\\\ /\ (0)(1)(2)(3) ///\\/\\ /\ (0)(1)(2)
depth 0123 0 model= 2 1 1 1 012 1 0 model= 2 2 1
/\
//\\ /\ depth /\ /\ depth
///\\\ //\\ (0)(1)(2) //\\ //\\ /\ (0)(1)
depth 012 01 model= 2 2 1 01 01 0 model= 3 2
/\
/ \ depth
//\/\\ /\ /\ (0)(1)
depth 01 1 0 0 model= 3 2
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.
T(3,1) = 2 different models= 2 2 1 and 2 1 1 1;
T(3,2) = 1 model= 3 2 (End).
LINKS
Indranil Ghosh, Rows 0..100, flattened
FORMULA
T(n,k) = binomial(n-k-1, ceiling(n/2)-1) if 0 <= k <= floor(n/2).
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.
EXAMPLE
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).
Triangle starts:
1:
1;
1, 1;
2, 1;
3, 2, 1;
6, 3, 1;
10, 6, 3, 1;
MAPLE
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
MATHEMATICA
Flatten[Table[Binomial[n-k-1, Ceiling[(n/2)-1]], {n, 0, 16}, {k, 0, Floor[n/2]}]] (* Indranil Ghosh, Mar 05 2017 *)
PROG
(PARI)
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(); ); });
tabf(16); \\ Indranil Ghosh, Mar 05 2017
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 06 2011
STATUS
approved