A215796 Number of distinct values taken by 7th 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, 283, 691, 1681, 3988, 9241, 20681, 44217, 89644

Offset: 1

Author

Alois P. Heinz, Aug 24 2012

Keywords

• nonn
• more

Examples

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 7th derivative at x=1: (x^(x^(x^x))) -> 26054; ((x^x)^(x^x)), ((x^(x^x))^x) -> 41090; (x^((x^x)^x)) -> 47110; (((x^x)^x)^x) -> 70098.
		

Crossrefs

Column k=7 of A216368.

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, A215837.

Programs

Maple

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
    end():
a:= n-> nops({map(f-> 7!*coeff(series(subs(x=x+1, f), x, 8), x, 7), T(n))[]}):
seq(a(n), n=1..12);