OFFSET
1,3
EXAMPLE
a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.
MAPLE
f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0, 0]},
{seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
+15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[4], f(n))):
seq(a(n), n=1..20);
MATHEMATICA
f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length@Union@(#[[4]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)
CROSSREFS
Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215835. Column k=5 of A216368.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 04 2011
STATUS
approved