[seqfan] Triangular combinations
hv at crypt.org
hv at crypt.org
Wed Jun 21 18:28:37 CEST 2023
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
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