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