I'd like some help

[Emeric Deutsch]

Dear seqfans,

I have defined a sequence of polynomials Q[n](t,s,x) by the recurrence 
relation
Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + tQ[n-1] if n is odd and

Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + sQ[n-1] if n is even;

Q[0]=1.

I would like to find the tetravariate e.g.f

G(t,s,x,z) = Q[0] + Q[1]*z/1! + Q[2]*z^2/2! + ...

It seems that, if not for the n-odd, n-even bifurcation, the
problem is routine: find the pde satisfied by G (apparently,
linear and homogeneous), associate the system of ode
dt/A=ds/B=dx/C=dz/F, and so on. I would do it by hand.

But in this case I am too lazy to try it by hand, especially
that I am afraid that it is not routine anymore. I wonder if
some program wouldn't be able to do the job.

Many thanks for any input.

I have obtained several new sequences with the above rec. 
relation. But the knowledge of the g.f. would make their
presentation much simpler and more complete.

For those interested let me give some details.

I am considering partitions of the set {1,2,...,n} and looking
for the statistics:
  number of odd blocks (blocks with all entries odd), marked by t;
  number of even blocks (blocks with all entries even), marked by s;
  number of mixed blocks (blocks containing both odd and even entries), 
marked by x. Then one can easily show that the generating polynomials 
Q[n](t,s,x) of the partitions of {1,2,...,n} are given by the above
rec. relation (requests for details are welcome).

For example, we find Q[3] = ts + 2tx + x + st^2, the terms
corresponding to 13|2, 1|23, 12|3, 123, and 1|2|3, respectively.

With thanks,
Emeric