Number of partitions of an integer with extra conditions In how many ways can you write number 15 as a sum of positive integers if number 1 can appear 3 times at most in the sum and number 3 can only appear even number of times?
I've tried doing this as $$(1+t+t^2+t^3)(1+t^2+t^4+t^6+t^8+t^{10}+t^{12}+t^{14})(1+t^6+t^{12})\cdots(1+t^{15})$$ and finding the coefficient of $t^{15}$, but it's taking too much time. Is there an easier way to do this? I think the result should be around 81. (I found the number of partitions of number 15 with no limitations, then ruled out those that do not satisfy the condition.)

Via inclusion-exclusion, where $p_n$ is the number of unrestricted partitions of $n$:
\begin{align}
& p_{15} 
- \left[p_{15-4\cdot1} + (p_{15-1\cdot3} - p_{15-2\cdot3}) + (p_{15-3\cdot3} - p_{15-4\cdot3}) + p_{15-5\cdot3} \right]
+ \left[(p_{15-4\cdot1-1\cdot3} - p_{15-4\cdot1-2\cdot3}) + p_{15-4\cdot1-3\cdot3} \right] \\ &= p_{15} 
- \left[p_{11} + (p_{12} - p_9) + (p_6 - p_3) + p_{0} \right]
+ \left[(p_8 - p_5) + p_{2}\right] \\
&= 176 
- \left[56 + (77 - 30) + (11 - 3) + 1 \right]
+ \left[(22 - 7) + 2\right] \\
&= 176 - 112 + 17 \\
&= 81
\end{align}
