Is there combinatorial recurrence with radicals?

Question

Is there recurrence relation for combinatorial objects which have root on right-hand side? I found that formula for problem in this question is of such type and want to find similar formulas and methods to obtain them and corresponding area of combinatorics. Example of triangle of numbers (from simple variant of mentioned question):

    [1]    
    [1, 4]    
    [1, 9, 16]    
    [1, 16, 49, 64]    
    [1, 25, 121, 225, 256]    
    [1, 36, 256, 676, 961, 1024]    

In it $A[x][y] = A[x-1][y-1] + A[x-1][y] + 2*\sqrt{A[x-1][y-1] * A[x-1][y]}$

Answer 1

The numbers $A[x,y]$ are the squares of the number of OEIS sequence A008949 which has the same recursion as Pascal's triangle for binomial coefficients. So, if you let $A[x,y]:=T[x,y]^2$, then $T[x,y]=T[x-1,y-1]+T[x-1,y]$ and the "radical" disappers. We can rewrite the recursion as $A[x,y]=A[x-1,y-1]+A[x-1,y]+2T[x-1,y-1]T[x-1,y]$ using $T[,]$.

This does not answer your question, but it is unlikely that the recurrences you want exist.