Fw: improvisation on Royal Paths

[wouter meeussen]

future users could possibly profit from an explicit definition of 'Royal
Paths'
using the Schroeder Triangle:  A033877 has everything already, but more
cross refs
would brighten it up :

T[1,_]:=1;
T[n_,k_]/;(k<n) :=0;
T[n_,k_]:=T[n,k]=T[n,k-1]+T[n-1,k-1]+T[n-1,k];

A006318= main diagonal of A033877
    Table[T[n,n],{n,8}]    {1, 2, 6, 22, 90, 394, 1806, 8558}
A006319= second diagonal of A033877
    Table[T[n,n+1],{n,8}]  {1, 4, 16, 68, 304, 1412, 6752, 33028}
A006320= third diagonal of A033877
    Table[T[n,n+2],{n,8}]  {1, 6, 30, 146, 714, 3534, 17718, 89898}
A006321= fourth diagonal of A033877
    Table[T[n,n+3],{n,8}]  {1, 8, 48, 264, 1408, 7432, 39152, 206600}

cherry on the cake:
A001003 = row sums of A033877.


Wouter.