[seqfan] Triangular combinations

[hv at crypt.org]

I'm writing comb(n, r) for nCr = n!/(r! (n - r)!).

This triangle arose in a problem I've been investigating. I'm always
unsure about adding 2-variable functions to OEIS, should it be added?

The numbers also look like they should have a simpler definition, but
I've not yet been able to see it.

  a(m, n) = sum_{k=0}^inf { comb(m, k) comb(m + n - k, k) }

Going by antidiagonals, I get:

m\n | 0 1  2  3  4  5   6
----+------------------
0   | 1
1   | 1 1
2   | 1 2  3
3   | 1 3  5  7
4   | 1 4  8 13 19
5   | 1 5 12 22 35 51
6   | 1 6 17 35 61 96 141

I think this can be interpreted roughly as: given a string of m+n letters
from an alphabet { a, b, c }, and given that 'a' does not appear in the
first m letters, count the number of possible strings in which a and b
occur equally often (possibly zero times).

It actually arose counting squares in a hypercube, on which more later.

Hugo