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Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able (after lots of routine algebraic simplification and reorganization) to rewrite this as \begin{align*} \left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j} \\ &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2} \frac{(n - j - 1)(j^2 + 4jn + n^2 + 5j + 7n + 6)}6\, t^{n+j} \end{align*} where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able (after lots of routine algebraic simplification and reorganization) to rewrite the right-hand side as \begin{align*} \left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j} \\ &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2} \frac{(n - j - 1)(j^2 + 4jn + n^2 + 5j + 7n + 6)}6\, t^{n+j} \end{align*} where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able (after lots of routine algebraic simplification and reorganization) to rewrite this as $$\left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 =\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j}$$ where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able (after lots of routine algebraic simplification and reorganization) to rewrite this as \begin{align*} \left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j} \\ &=\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2} \frac{(n - j - 1)(j^2 + 4jn + n^2 + 5j + 7n + 6)}6\, t^{n+j} \end{align*} where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able to rewrite (after lots of routine algebraic simplification and reorganization) this as $$\left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 =\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j}$$ where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write $$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$ Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form $$(nt^{n+1}-(n+1)t^n+1)^2=(1-t)^3[(1-t^n)b_{n-1}-t(1-t^{n+1})b_{n-2}].$$ Now, as Mark Widon mentioned, try to read-off the coefficients of $t^k$.

UPDATE. Resorting back the original formulation of the claim $$a_n^2=b_{n-1}\sum_{j=0}^{n-1}t^j-t\,b_{n-2}\sum_{j=0}^nt^j,$$ I was able (after lots of routine algebraic simplification and reorganization) to rewrite this as $$\left(\sum_{j=0}^{n-1}(j+1)t^j\right)^2 =\sum_{j=0}^{n-1}\binom{3+j}3t^j+\sum_{j=0}^{n-2}\left[\beta_n-\beta_{j+1}-\binom{n-j}3\right]\,t^{n+j}$$ where $\beta_k=\frac{k(k+1)(2k+1)}6$ (the sum of squares function).

Once we got this far, the next step is to compare the coefficients of $t^k$.