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## Main Question or Discussion Point

I am looking for a solution to a multivariable recursive formula as follows:

Define:

Initial Conditions:

a(1,0)=-3, a(2,0)=2, a(n,0)=0 for n≠1,2

Recursive Formula:

a(n,k)=(4n+1)a(n,k-1)-4a(n-1,k-1)

Find a formula in terms of n and k for a(n,k).

This is my first post and so I may have chosen the wrong category. This seems to be a differential equation problem since most single variable recursive formulas are solved using some series or differential equations. For this problem I am quite stuck. I have already tried to use (possibly incorrectly) mathematica to solve it for me, but the program failed to solve it.

If anyone has any insight or knowledge on solving general recursive equations, I would be most appreciative. Even a proof that no such function exists would be really helpful.

Define:

Initial Conditions:

a(1,0)=-3, a(2,0)=2, a(n,0)=0 for n≠1,2

Recursive Formula:

a(n,k)=(4n+1)a(n,k-1)-4a(n-1,k-1)

Find a formula in terms of n and k for a(n,k).

This is my first post and so I may have chosen the wrong category. This seems to be a differential equation problem since most single variable recursive formulas are solved using some series or differential equations. For this problem I am quite stuck. I have already tried to use (possibly incorrectly) mathematica to solve it for me, but the program failed to solve it.

If anyone has any insight or knowledge on solving general recursive equations, I would be most appreciative. Even a proof that no such function exists would be really helpful.