[seqfan] Filling a triangle with numbers

[Neil Sloane]

I heard this problem in Hungary recently:

How many ways are there to arrange the numbers 1 through n(n+1)/2 in
a triangle so that each term is between the two numbers
immediately below it (and each row is increasing)?

Examples:
.2.
1.3

..4..
.2.5.
1.3.6

..3..
.2.5.
1.4.6

I get a(1)=a(2)=1, a(3)=2, a(4)=6?

The only related sequences I can find in the OEIS are A064049, A064059,
which are different. However, it is quite likely this is a well-studied
problem.

Also, what if one omits the clause in parentheses in the definition?

Neil