Also:
ClearAll[toLTM]
toLTM[n_Integer] := Composition[Transpose, PadLeft, Range, Reverse, Range]@n
(* or toLTM[n_Integer] := Reverse /@ PadLeft @ Range @ Range @ n *)
toLTM[v_List] := Transpose @
PadLeft @ Extract[v, List /@ Range @ Reverse @ Range @ Length @ v]
toLTM @ 4 // TeXForm
$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 \\ 4 & 3 & 2 & 1 \\ \end{array} \right)$
toLTM @ CharacterRange["a", "d"] // TeXForm
$\left( \begin{array}{cccc} \text{a} & 0 & 0 & 0 \\ \text{b} & \text{a} & 0 & 0 \\ \text{c} & \text{b} & \text{a} & 0 \\ \text{d} & \text{c} & \text{b} & \text{a} \\ \end{array} \right)$
