I am trying to prove the following conjecture.
$$\sum_{k=0}^{n}(4k+3)^{m}\, T(n,k) = \sum_{k=0}^{n}(4(n-k)+1)^{m}\, T(n,k)$$ where $T(n,k)$ are a sequence of numbers (it isn't really relevant to know anything else about them), $n\in\mathbb{N}_{0}$ and $m$ is any natural number between $0$ and $n$ inclusive (it does not necessarily hold for higher $m$).
I have shown $$\sum_{k=0}^{0}(4k+3)^{m}\, T(n,k) = \sum_{k=0}^{0}(4(n-k)+1)^{m}\, T(n,k)$$ is true for $m=0$ (all that is required by the conjecture as $n=0$). And I have also shown $$\sum_{k=0}^{n+1}(4k+3)^{m}\, T(n+1,k) = \sum_{k=0}^{n+1}(4(n-k)+5)^{m}\, T(n+1,k)$$ under the assumption $$\sum_{k=0}^{n}(4k+3)^{r}\, T(n,k) = \sum_{k=0}^{n}(4(n-k)+1)^{r}\, T(n,k)$$ is true for $r\in \lbrace 0,1,\dots,m\rbrace$.
Is this a complete proof by induction? I'm not very settled on it as $m$ is dependant on $n$ and my induction hypothesis seems to be assuming more than I am allowed. Some guidance on which specific statements I should prove to have a complete proof of the theorem would be helpful.