If $k \leq n$ I denote the Young diagram with shape $(n,n-1,n-2,\ldots,1)$ by $\lambda^{n,n-1,\ldots,1}$. I write $f^{\lambda_n^{n,n-1,\ldots,1}}$ to count the number of semistandard Young tableaux with shape $\lambda^{n,n-1,\ldots,1}$ and maximum entry $n$. For example if $n=3$ then we draw: $\lambda^{3,2,1}= \begin{matrix} \bullet & \bullet & \bullet\\ \bullet & \bullet & \\ \bullet & & \\ \end{matrix}$ and it can be shown that $f^{\lambda_{3}^{3,2,1}}=8.$ I can construct a sequence for $f^{\lambda_n^{n,n-1,\ldots,1}}$ starting with $n=1$ and it is $$1,2,8,64,1024,3278,\ldots$$ Inspection show this sequence appears to be A006125 modulo an offset of +1. If $(i,j)$ is any cell in $\lambda^{n,n-1,\ldots,1}$ and $h(i,j)$ and $j-i$ are the hook lengths and content of the cell respectively then we can use Stanley's content formula and write
$$\prod_{(i,j) \in \lambda^{n,n-1,\ldots,1}}{n+j-i\above 1.5pt h(i,j)}=2^{n(n-1)\above 1.5pt 2} $$
Question: Let $\lambda^{n,n-1,\ldots,1}$ be a Young diagram with triangular shape and let $t_n={n(n-1)\above 1.5pt 2}.$ If $f^{\lambda_n^{n,n-1,\ldots,1}}$ counts the number of semistandard Young Tableaux of shape $\lambda^{n,n-1,\ldots,1}$ and maximum entry $n$ is it true $$f^{\lambda_n^{n,n-1,\ldots,1}}=2^{t_n}$$ ?