I am trying to identify a pattern in the following set of equations;
$$N_0=y$$ $$N_1=y^2+z$$ $$N_2=y^3+2yz$$ $$N_3=y^4+3y^2z+z^2$$ $$N_4=y^5+4y^3z+3yz^2$$ $$N_5=y^6+5y^4z+6y^2 z^2+z^3$$
Essentially, I need to be able to produce an equation for $N_h$. I've been staring at it for quite some time now, and have been unable to identify the pattern. Any assistance would be greatly appreciated. Is anybody aware of a particular methodology which could be applied here?
For each fixed power of $z$, the coefficients of the increasing powers of $y$ seem to be the diagonals of Pascal's triangle. For instance, the coefficients of $z^2$ seem to be $\binom22=1$, $\binom32y=3y$, $\binom42y^2=6y^2$; since they start at $k=3$, the general pattern is $\binom{k-1}2y^{k-3}$ (which works even when $k<3$). The coefficients of $z$ seem to be $\binom11=1$, $\binom21y=2y$, $\binom31y^2=3y^2$, and so on; since they start at $k=1$, the general pattern is $\binom k1y^{k-1}$. Even the pure powers of $y$ can be written as $\binom{k+1}0y^{k+1}$, which is consistent with the other two patterns. (It thus seems natural to write $N_{-1}=1$; perhaps the whole sequence's indices should be shifted by $1$.)
The general formula could then be written as $$ N_k = \sum_{j=0}^\infty \binom{k+1-j}j y^{k+1-2j} z^j, $$ where the series is really finite, stopping at $j=\lceil\frac k2\rceil$.