How many distinct distributions (x,y,z,w) of n
identical marbles in 4 boxes labeled A,B,C and D are there, such that

x,y,z,w are positive integers in strictly increasing order?

Verify the validity of your formula (or set of formulas) by manual listing of all such distributions for n=18.

(In reply to

solution by Charlie)

Taking this sequence 1,1,2,3,5,6,9,... and seaching the OEIS yields A001400 "Number of partitions of n into at most 4 parts." In the comments it is noted that this sequence is the "Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0<x<y<z<p."