closed form for A053495

[Wouter Meeussen]

A053495

combining the even & odd diagonals as
f[r_,c_]:=c! Pochhammer[c-r+1,r]/(r!)^2
g[r_,c_]:=c! Binomial[c-1,r]/(r+1)!
Table[If[EvenQ[r + c],  f[(r - c)/2, (r + c)/2],  g[(r - c - 1)/2,  (r + c + 1)/2]], {r, 0, 7}, {c, 0, r}]

gives a closed form:
      Table[
(-1)^(r+c+1) Binomial[Floor[(r+c)/2], Floor[(r-c)/2]] Floor[(r+c+1)/2]! / Floor[(r-c+1)/2]! 
     ,{r,0,7},{c,0,r}]

amazing what a simple recurrence like 
Clear[a];a[0]:=-1;a[1]:=1-x;a[n_]:=a[n]=a[n-2]+n x a[n-1]
can lead to. 
Can anyone transform the above recurrence into a GF?

ps. could anyone please teach me about those Laguerre polynomials in 
A001809, A001810 and
A001811=Table[n!*n*(n-1)(n-2)(n-3)/(4!)^2,{n,1,12}]

Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
wouter.meeussen at vandemoortele.com