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 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.
MAPLE
f:= proc(n) option remember;
`if`(n=1, {[0, 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],
45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+
270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+
60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+
100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[5], f(n))):
seq(a(n), n=1..15);
CROSSREFS
Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215836. Column k=6 of A216368.
KEYWORD
nonn,more
AUTHOR
Alois P. Heinz, Nov 11 2011
EXTENSIONS
a(22)-a(23) from Alois P. Heinz, Sep 26 2014
STATUS
approved