How to draw an arrow with a label head inside a tabular?

You are misusing the table with extra coloumn and row to make the arrows longer. If you want the arrows parallel and end in a nice way, tikzmark does not make sense as it draws the arrows relative to the numbers or in your case the extra empty cells

This looks terrible:

documentclass[preview, border=1cm]{standalone}
usepackage{tabularx}
usepackage{tikz} 
usetikzlibrary{tikzmark}

renewcommandtabularxcolumn[1]{m{#1}}
newcolumntype{M}{>{centeringarraybackslash}m{1cm}}

newcommandlink[3]{%
begin{tikzpicture}[remember picture, overlay, >=stealth, shift={(0,0)}]
  draw[->] (pic cs:#1) -- (pic cs:#2) node[above right]{#3};
end{tikzpicture}%
}

begin{document}

noindent
begin{tabularx}{textwidth}{MMMMMMMMMX}
               &         &tikzmark{y}{} &               &     &   &&&&[2em] %cline{1-6}
               &1        &tikzmark{b}{} &tikzmark{d}{} &     &     &&&&[2em]  
tikzmark{x}{} &1        &1              &tikzmark{f}{} &tikzmark{h}{}&  &&&&[2em]  
tikzmark{a}{} &1        &2              &1              &tikzmark{j}{}&tikzmark{l}{} &&&&[2em]  
tikzmark{c}{} &1        &3              &3              &  1&  &&&&[2em]  
tikzmark{e}{} &1        &4              &6              &4&1&&&  [2em]
tikzmark{g}{} &1        &5              &10             &10&5&1&&  [2em]
tikzmark{i}{} &1        &6              &15             &20&15&6&1&  [2em]
tikzmark{k}{} &$vdots$ &$vdots$       &$vdots$       & $vdots$ & $vdots$ & $vdots$ & $vdots$ & $ddots$
end{tabularx}
link{x}{y}{1}
link{a}{b}{1}
link{c}{d}{2}
link{e}{f}{3}
link{g}{h}{5}
link{i}{j}{8}
link{k}{l}{13}
end{document} 

To rotate the sums as in your drawing, you can use this:

draw[->] (pic cs:#1) -- (pic cs:#2) node[above right, rotate=45]{#3};  

This is a bit different, but why don't you use a matrix? For example:

documentclass[preview, border=1cm]{standalone}
usepackage{tikz}
usetikzlibrary{tikzmark, matrix, calc, arrows.meta, backgrounds}

begin{document}
begin{tikzpicture}[myline/.style={
        orange!20, line width=1.2em, cap=round, -Triangle Cap},]
        matrix (m) [matrix of nodes, nodes in empty cells,
        nodes = {circle, minimum width=3em, anchor=center},
        ]
        {
            1 &   &    &    &    &   &
            1 & 1 &    &    &    &   &
            1 & 2 & 1  &    &    &   &
            1 & 3 & 3  & 1  &    &   &
            1 & 4 & 6  & 4  & 1  &   &
            1 & 5 & 10 & 10 & 5  & 1 &
            1 & 6 & 15 & 20 & 15 & 6 & 1
        };
        begin{scope}[on background layer]
            draw [myline] (m-1-1.center) -- (m-1-1.center) -- ++(45:1) coordinate(t1);
            draw [myline] (m-2-1.center) -- ($(m-1-1.center)!0.5!(m-2-2.center)$) -- ++(45:1) coordinate(t2);
            draw [myline] (m-3-1.center) -- (m-2-2.center) -- ++(45:1) coordinate(t3);
            draw [myline] (m-4-1.center) -- ($(m-2-2.center)!0.5!(m-3-3.center)$) -- ++(45:1) coordinate(t4);
            draw [myline] (m-5-1.center) -- (m-3-3.center) -- ++(45:1) coordinate(t5);
            draw [myline] (m-6-1.center) -- ($(m-3-3.center)!0.5!(m-4-4.center)$) -- ++(45:1) coordinate(t6);
            draw [myline] (m-7-1.center) -- (m-4-4.center) -- ++(45:1) coordinate(t7);
        end{scope}
        foreach [count=i] tot in {1, 1, 2, 3, 5, 8, 13} node[font=bfseries] at (ti) {tot}; 
end{tikzpicture}
end{document}