Whatever this is, it's a cute counting argument. You begin with Pascal's triangle and observe its symmetry (to abuse all kinds of notation, any nxn square matrix subset of Pascal's triangle is equal to its tranpose). This square "row/column" treatment of the triangle is a rephrase of the symmetry.
Please explain exactly what you mean by "l" and "d" in the middle bit. Are you referring to individual entries, partial (sums of) rows/columns, etc.
Just as in the triangle itself, a quick check of your latter polynomials shows one pattern in the second-to last terms: add powers of 2 to get the next one. 2 + 4 = 6, 6 + 8 = 14, 14 + 16 = 30, 30 + 32 = 62... And the leading terms are just the factorials.
