from 2^n to catalan(n), the hard way 'round

[wouter meeussen]

the substitution game
i_Integer -> Range[0, Mod[i+1, w, 1] starting on 0
yields a different sequence for each w.
(* in Mathematica, the third argument is an offset,
so that Mod[5,5,1]=5 and Mod[6,5,1]=1 *)

For w-> Infinity, this is just
i_Integer -> Range[0, i+1] starting on 0
yielding the catalans (A000108)

thanks to Dion Gijswijt and A011973,
I could find the set of OGF's parametrised for w:

Table[
Sum[-(-1)^(k) Binomial[w-k,k] x^k, {k,0,Floor[w/2]}]/
(x^(w+1) +
 Sum[-(-1)^(k) Binomial[2+w-k,k] x^k, {k,0,Floor[1+w/2]}])
,{w,0,15}]


So, the limit for w-> Infinity of the above is
-1/x +2/x /(1+Sqrt[1-4 x])

For the substitution game
i_Integer -> Range[0, Min[i+1, w] starting on 0
this is alreasy in EIS, under A080938,
with comment "Cf. A000007, A000012, A011782, A001519, A007051,
 A080937, A024175, A080938, A033191 which essentially provide
the same sequence for different limits, and tend to A000108."

I can only add the parametrised OGF :

Table[
Sum[-(-1)^(k)Binomial[w-k,k]x^k,{k,0,Floor[w/2]}]/
Sum[-(-1)^(k)Binomial[2+w-k,k]x^k,{k,0,Floor[1+w/2]}]
,{w,0,12}]

with the same limit as above.

W.