Young Tableaux

[Emeric Deutsch (RP)[M}]

Hi,
By sheer coincidence, I am reading a paper where the
sequence A000085 (1,1,2,4,10,26,76, ...) comes up 
(Duluck and Penaud; Discrete Math., 117, 1993, 89-105).  
Defined as number of involutions (i. e. self-inverse 
permutations) on n letters, one can easily derive

                         floor(1/2 n)
                            -----
                             \               n!
               J := n ->      )       ----------------
                             /                       k
                            -----     k! (n - 2 k)! 2
                            k = 0
(authors refer to Lucas' Theorie des Nombres, Vol. 1).
Indeed, Maple gives
  1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504,
        2390480, 10349536, 46206736, 211799312, 997313824, 4809701440,

I couldn't resist the simple exercise of deriving the ordinary 
generating function G(z). It is defined by
> deq:=z^3*diff(G(z),z)-(1-z-z^2)*G(z)+1 = 0;

                     3 /d      \             2
             deq := z  |-- G(z)| - (1 - z - z ) G(z) + 1 = 0
                       \dz     /

                            ic := G(0) = 1
Maple gives the solution I couldn't decipher:
                                            1
            sol := G(z) = signum(exp(- 1/2 ----)) infinity
                                             2
                                            z
However, with the "series" command
> sol:=dsolve({deq,ic},G(z),series);
we obtain
                           2      3       4       5       6        7
  sol := G(z) = 1 + z + 2 z  + 4 z  + 10 z  + 26 z  + 76 z  + 232 z

                8         9         10          11           12
         + 764 z  + 2620 z  + 9496 z   + 35696 z   + 140152 z   +

                13            14             15             16
        568504 z   + 2390480 z   + 10349536 z   + 46206736 z   +

                   17              18               19      20
        211799312 z   + 997313824 z   + 4809701440 z   + O(z  )

Emeric