OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(x)$ with coefficients $0$ or $1$ and degree at most $n$, it denotes by $T(n,k)$, the number of such polynomials of thickness $k$. The numbers $T(n,k)$ form the triangle \begin{align*} &[1, 1]\\ &[1, 2, 1]\\ &[1, 3, 3, 1]\\ &[1, 4, 8, 2, 1]\\ &[1, 5, 16, 6, 3, 1]\\ &[1, 6, 29, 14, 11, 2, 1]\\ &[1, 7, 49, 29, 29, 9, 3, 1]\\ &[1, 8, 82, 52, 74, 22, 14, 2, 1]\\ &[1, 9, 130, 96, 160, 58, 42, 12, 3, 1]\\ &[1, 10, 205, 160, 344, 128, 126, 30, 17, 2, 1]\\ &[1, 11, 305, 277, 676, 294, 314, 98, 53, 15, 3, 1]\\ &[1, 12, 450, 450, 1333, 576, 796, 232, 185, 38, 20, 2, 1]\\ &[1, 13, 654, 712, 2477, 1177, 1796, 628, 501, 147, 64, 18, 3, 1]\\ &[1, 14, 947, 1086, 4563, 2212, 4075, 1370, 1425, 368, 251, 46, 23, 2, 1]\\ &[1, 15, 1343, 1657, 7997, 4289, 8535, 3265, 3515, 1117, 729, 205, 75, 21, 3, 1]\\ &\cdot\\ &\cdot\\ &\cdot \end{align*} which is elaborated in the sequence mentioned before.
My question is whether the concept of thickness of a polynomial has been studied anywhere else in more details. In particular, are the numbers $T(n,k)$ well known? Do we have formulas for these numbers (exact or recursive)? If not, then are there any asymptotic results? Or even any interesting bounds, or other interesting properties?
Of course, some properties are quite obvious - for example, $T(n,0)=T(n,n+1)=1$ and $T(n,1)=n+1$. And some others seem to have a pattern although they are not easy to explain - for example, $T(n,n)$ seems to follow the pattern $(2,3,2,3,\dots)$ and $T(n,n-1)$ seems to increase by $5$ and decrease by $2$ every second step and goes $(3,8,6,11,9,14,\dots)$. I can't even seem to explain where these patterns are coming from. Also, what are the other non-trivial properties of this triangle?
I wrote an email to Prof. Sloane asking about the origin of these numbers and he replied that the concept of thickness arose from studying numbers that have no carries when they are squared (skinny numbers). He also said that there are a lot of entries from 2010 that deal with thickness of polynomials, but nothing was published anywhere. So the OEIS seems to be the best reference to use so far.
Also posted on MathOverflow
