What is the probability that the numbers that come up will be in ascending order (not necessarily strictly ascending, but never a lower number after a higher)?

Intended to be solved without a computer.

1. How did you arrive at the sequence a(1) = 6; a(2) = 21; etc.

I trust it was adequately explained in my previous
comment.

2. Was the source Sloane's OEIS?
No, I Just looked up a DIAGONAL in Pascal triangle
and from the locations where 1,6,21,56.126,252 etc
appeared deduced that there are 
a(n)=C(n+k-1,n)  ascending sequences, n being number of throws  using a k-sided dice.
Then i've validated the formula for k=1 ==> a(n)=1 and for  n=4 k=3 listed all 15 outcomes.
Following your enquiry I visited SLOANE & found similar,
-not exactLY THE SAME= formula with another offset, did not bother to study it since no recursion or generating function was presented there.

3. Was a computer used in looking this up?

Intended to be solved without a computer.

and secondo , I always prefer the BRAIN, i.e. analytical approach.

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