Shallow diagonals of triangular tables

[Wouter Meeussen]

hi all,

the EIS contains a number of tiangular tables like Pascal's triangle.
Sometimes, the shallow diagonal sums have obvious special properties,
sometimes not.

example:

                                1
                              1   1
                            1   2   1
                          1   3  +3  (1)
                        1  +4  (6)  4   1
                     +1  (5)  10  10  5   1  
                   (1)

gives shallow diagonal 1,1,2,3,5,+8,(13),21,... the Fibonacci numbers.

An other triangular table, a favorite of mine, the Catalan triangle, has as
shallow diagonal sum the central binomial coefficients Binomial[n,Floor[n/2]] :

                                 1
                               1,  1
                             1,  2,  2
                           1,  3,  5, (5)
                         1,  4, (9)  14, 14
                       1, (5)  14,  28,  42, 42
                    (1)  6,  20,  48,  90, 132, 132


shallow diagonal sum : 1,1,2,3,6,10,(20),35,70,126,252 ...
Since for both these triangles, the non-recursive form of each entry is known
(* Pascal's : Binomial[n,m] ; Catalan's : Binomial[n+m,n](n-m+1)/(n+1) ; *)
thus also the shallow diagonal has a simple form;

in Mathematica, it is Sum[ triangle[n-k,k-1],{k,1,Floor[(n+1)/2]}]
with triangle[n,k] the n-th row and the k-th column in the triangle.

The "opposite" sloping shallow diagonal (top left to bottom right) has the form:
Sum[triangle[n-k,(n-k)-(k-1)],{k,1,Floor[(n+1)/2]}]
and for a non-symmetrical triangle like Catalan's it is of course different:
 1,1,3,7,20,59,184,593,1964,6642,22845,79667,281037,1001092,3595865 ...
but this does not appear to "count" any known combinatorial objects.

QUESTION:

If a triangular table has both a recurrence relation and a closed form,
and is related to the counting of some combinatorial type of object,
does this imply that the "Shallow Diagonals" have a combinatorial significance?

In other words, are the simple forms obtained for Pascal's and Catalan's
triangle just coincidence ? What about the other triangular tables in EIS?

wouter.



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