A216062 Number of distinct values taken by 9th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1838, 4734, 12247, 31617, 81208

Offset: 1

Author

Alois P. Heinz, Aug 31 2012

Keywords

• nonn
• more

Comments

a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 9th derivative at x=1: (x^(x^(x^x))) -> 3010680; ((x^x)^(x^x)), ((x^(x^x))^x) -> 3863808; (x^((x^x)^x)) -> 6019416; (((x^x)^x)^x) -> 6333336.

Crossrefs

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215839. Column k=9 of A216368.

Programs

Maple

# load programs from A215703, then:
a:= n-> nops({map(f-> 9!*coeff(series(subs(x=x+1, f),
                  x, 10), x, 9), T(n))[]}):
seq(a(n), n=1..11);