How to generalize a formula?

Question

This is a piece of my code:

der[g_, x_, n_] := Sum[g[k, g[0, x]] BellY[n, k, 
         Table[g[i, x], {i, n - k + 1}]], {k, 0, n}]
g[0, x] := x
g[1, x] := x
g[2, x_] := Evaluate[der[g, x, 2 - 1]]
g[3, x_] := Evaluate[der[g, x, 3 - 1]]
num := 4
Expand[der[g, x, num]]

It works up to n=4. I want to make it work for higher n.

But trying to define

    g[n_, x_] := Evaluate[der[g, x, n - 1]]

returns error. What should I do?

Answer 1

I don't have any intuition whatsoever as to what your expected output is. The error message that emerges can be dealt with if you increase your iteration and recursion limits but seeing as I don't really know exactly how BellY works I leave this to you. If you write your g function recursively, like so

der[g_, x_, n_] := 
 Sum[g[k, g[0, x]] BellY[n, k, Table[g[i, x], {i, n - k + 1}]], {k, 0,
n}]
g[1, x] = x;
g[n_, x_] := g[n, x] = der[g, x, n - 1]

and evaluate:

Expand[der[g, x, 4]]

you get

x^5 + 11 x^6 + 11 x^7 + 8 x^8 + 4 x^9 + x^10 + x^11

And something more complicated, like evaluating:

Expand[der[g, x, 13]]

outputs (after some warnings re low recursion limits):

x^14 + 8178 x^15 + 1479726 x^16 + 45923846 x^17 + 459029454 x^18 + 
2208858006 x^19 + 7075642044 x^20 + 16806933760 x^21 + 
31816904623 x^22 + 51866622093 x^23 + 74193541415 x^24 + 
98180859588 x^25 + 119082181277 x^26 + 137536650290 x^27 + 
150143774552 x^28 + 157992149040 x^29 + 160050416326 x^30 + 
157096816119 x^31 + 150189370525 x^32 + 139814563126 x^33 + 
127454419592 x^34 + 113603426826 x^35 + 99578396788 x^36 + 
85494865250 x^37 + 72430802460 x^38 + 60247338445 x^39 + 
49525049687 x^40 + 40031803598 x^41 + 32046236578 x^42 + 
25240371714 x^43 + 19719393876 x^44 + 15171635174 x^45 + 
11589845200 x^46 + 8724377910 x^47 + 6528035076 x^48 + 
4813974664 x^49 + 3532331160 x^50 + 2554783500 x^51 + 
1840197183 x^52 + 1306568236 x^53 + 924434043 x^54 + 
644645508 x^55 + 448470745 x^56 + 307216681 x^57 + 210203601 x^58 + 
141532130 x^59 + 95259446 x^60 + 63040614 x^61 + 41765538 x^62 + 
27144679 x^63 + 17701948 x^64 + 11305600 x^65 + 7250417 x^66 + 
4546412 x^67 + 2871683 x^68 + 1763322 x^69 + 1096094 x^70 + 
660630 x^71 + 402853 x^72 + 237206 x^73 + 142759 x^74 + 81694 x^75 + 
48320 x^76 + 26994 x^77 + 15606 x^78 + 8419 x^79 + 4842 x^80 + 
2476 x^81 + 1395 x^82 + 694 x^83 + 374 x^84 + 173 x^85 + 98 x^86 + 
38 x^87 + 22 x^88 + 8 x^89 + 4 x^90 + x^91 + x^92

Again, with the caveat that I don't know the specific polynomial expansion so it may well be that your expression for der doesn't always converge.