|
| |
|
|
A199205
|
|
Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
|
|
10
|
|
|
|
1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
LINKS
|
Table of n, a(n) for n=1..40.
|
|
|
EXAMPLE
|
a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.
|
|
|
MAPLE
|
f:= proc(n) option remember;
`if`(n=1, {[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]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops (map (x-> x[3], f(n))):
seq (a(n), n=1..20);
|
|
|
CROSSREFS
|
Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.
Sequence in context: A207813 A136379 A065026 * A192967 A182806 A007502
Adjacent sequences: A199202 A199203 A199204 * A199206 A199207 A199208
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Alois P. Heinz, Nov 03 2011
|
|
|
STATUS
|
approved
|
| |
|
|
