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I would like to find the general expression for a recursive equation:

RSolve[{a[m, n, p] - 3 a[m - 1, n, p] - a[m, n - 1, p] - 
    a[m, n, p - 1] == 0}, a[m, n, p], {m, n, p}]

with initial value $a[0,0,0]==1$

I have no idea how this can be done in MMA. Does anyone know how to do it?

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Making

a[m_, n_, p_] := alpha^n*beta^m*gamma^p
(a[m, n, p] - 3 a[m - 1, n, p] - a[m, n - 1, p] - a[m, n, p - 1])/a[m, n, p] // FullSimplify

gives as a solution all $\alpha,\beta,\gamma$ obeying the relationship

1 - 1/alpha - 3/beta - 1/gamma == 0

or $a(n,m,p) = \alpha^m\beta^n\gamma^p\ \ \mbox{s. t. }\ \ \frac{1}{\alpha}+\frac{3}{\beta}+\frac{1}{\gamma} = 1$

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