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Function fits with combinations


Hello,
I've been attempting to use Wolfram to help me identify a function but
after playing with "fit" it keeps giving me linear/quadratic/etc.
solutions and I am sure that the function generating my data is using
combinations. I have a series of n length/m length pairs and I've been
generating alignments between the two, and in the first segment of the
pair any character can map to 1-3 chars of the latter. The number of
alignments is as follows:

3,3 = 1
3,4 = 3
3,5 = 6
3,6 = 7
3,7 = 6
3,8 = 3
3,9 = 1

4,4 = 1
4,5 = 4
4,6 = 10
4,7 = 16
4,8 = 19
4,9 = 16
4,10 = 10
4,11 = 4
4,12 = 1

5,5 = 1
5,6 = 5
5,7 = 15
5,8 = 30
5,9 = 45
5,10 = 51
5,11 = 45
5,12 = 30
5,13 = 15
5,14 = 5
5,15 = 1

Can anyone help me to either identify the function in question or
figure out how to point to Wolfram that it almost certainly has to do
with combinations?

Thanks,
Richard



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