Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version): $$ T(0,0) = 2;T(1,0) = 1;T(1,1) = 1; $$ $$T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-2), n>1 $$

The sequence in question is generated by taking the largest element from each row, then dividing it by the row number $n$. The resulting sequence is as follows (listed up to $a_{20}$):

$1, 1.5, 2, 3, 5, 10, 19, 37, 77.\overline 7, 158, 331, 715, 1505, 3287, 7224.4, 15529, 34839, 77453.8 \overline3, 168950, 386050, ...$

My question is: **Why are the majority of these numbers convoluted convolved Fibonacci numbers?** Even the ones which are not integers become convoluted convolved Fibonacci numbers when rounded down to the nearest integer. Some examples:

$$floor(a_{15}) = 7224 = G_{11}^{(5)}$$ $$a_{16} = 15529 = G_{12}^{(5)}$$ $$a_{17} = 34839 = G_{12}^{(6)}$$ $$a_{19} = 168950 = G_{13}^{(7)}$$ $$a_{20} = 386050 = G_{14}^{(7)}$$ $$floor(a_{21}) = 865264 = G_{15}^{(7)}$$