Let $\mathcal A$ be a Banach algebra. If $n \in \mathbb N$ and $\mathcal I$ is a closed two sided ideal in $\mathcal A$, then $\mathcal A$ is $n−\mathcal I−$weakly amenable if the first cohomology group of $\mathcal A$ with coefficients in the $n−$th dual space $\mathcal I^{(n)}$ is zero, i.e., $H^1(\mathcal A, \mathcal I^{(n)} ) = \{0\}$. Further, $\mathcal A$ is $n−$ideally amenable (ideally amenable) if $\mathcal A$ is $n − \mathcal I−$weakly amenable $(1 − \mathcal I−$weakly amenable) for every closed two sided ideal $\mathcal I$ in $\mathcal A$. In this paper we investigate $(2m + 1) − \mathcal I−$weakly amenability of Banach algebras for $m \geq 1$, and ideal amenability of Segal algebras and triangular Banach algebras $T = \begin{bmatrix} \mathcal A & \mathcal M\\  &\mathcal B\end{bmatrix}$ (where $\mathcal A$ and $\mathcal B$ are Banach algebras and $\mathcal M$ is a $\mathcal A$, $\mathcal B−$module).