This is not exactly an answer, but it's too long for a comment and also probably contains all the information you need. From the form of the recursion, it looks like an enumeration of trees, set partitions, or something similar. It's easy to compute the first few rows of the array, and indeed, throwing the first few terms into the OEIS comes up with four compelling-looking hits:

http://www.oeis.org/A111672

http://www.oeis.org/~njas/sequences/A144150

http://www.oeis.org/~njas/sequences/A153277

(These appear to actually be three copies of essentially the same array that differ only in the number of 1s included at the boundary; there may also be other instances if one reads the array in a different order.) And I think that it should not be difficult to show that your numbers do indeed correspond to the numbers in any one of these tables.

This is not exactly an answer, but it's too long for a comment and also probably contains all the information you need. From the form of the recursion, it looks like an enumeration of trees, set partitions, or something similar. It's easy to compute the first few rows of the array, and indeed, throwing the first few terms into the OEIS comes up with four compelling-looking hits:

http://www.oeis.org/A111672

http://www.oeis.org/A144150

http://www.oeis.org/A153277

(These appear to actually be three copies of essentially the same array that differ only in the number of 1s included at the boundary; there may also be other instances if one reads the array in a different order.) And I think that it should not be difficult to show that your numbers do indeed correspond to the numbers in any one of these tables.

This is not exactly an answer, but it's too long for a comment and also probably contains all the information you need. From the form of the recursion, it looks like an enumeration of trees, set partitions, or something similar. It's easy to compute the first few rows of the array, and indeed, throwing the first few terms into the OEIS comes up with four compelling-looking hits:

http://www.research.att.com/~njas/sequences/A111672

http://www.research.att.com/~njas/sequences/A144150

http://www.research.att.com/~njas/sequences/A153277

(These appear to actually be three copies of essentially the same array that differ only in the number of 1s included at the boundary; there may also be other instances if one reads the array in a different order.) And I think that it should not be difficult to show that your numbers do indeed correspond to the numbers in any one of these tables.

This is not exactly an answer, but it's too long for a comment and also probably contains all the information you need. From the form of the recursion, it looks like an enumeration of trees, set partitions, or something similar. It's easy to compute the first few rows of the array, and indeed, throwing the first few terms into the OEIS comes up with four compelling-looking hits:

http://www.oeis.org/A111672

http://www.oeis.org/~njas/sequences/A144150

http://www.oeis.org/~njas/sequences/A153277

(These appear to actually be three copies of essentially the same array that differ only in the number of 1s included at the boundary; there may also be other instances if one reads the array in a different order.) And I think that it should not be difficult to show that your numbers do indeed correspond to the numbers in any one of these tables.