As is well known, the third coefficient is a triangle number. Looking at these numbers I believe the first coeeficient could be described as a point number and the second as a line number.
The fourth coefficient is the sum of consecutive triangle numbers. If these triangles are stacked on top of each other they form a tetrahedron. (I have a more clear rendition of this in the JPEG format. If you would like to see it, let me know via e-mail)
To me this suggests a sequence:
The first coefficient of a binomial expansion is a point number It is one point.
The second coefficient is a line number Lines are bound by 2 points
The third is a triangle number It is bound by 3 lines.
The fourth, a tetrahedron number. Tetrahedrons are bound by 4 triangles.
To me this suggests a sequence. Would the next coefficient be a number that forms an object in 4-space? Presumably this object would be bound by 5 tetrahedrons. The number after that would be a 5 space object bound by 6 whachamacallits, etc. etc.