A199883 Number of distinct values taken by 6th 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, 113, 262, 591, 1263, 2505, 4764, 8479, 14285, 22871, 35316, 52755, 76517, 107826, 148914, 202715, 270622

Offset

1,3

Links

Table of n, a(n) for n=1..23.

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.

Sequence in context: A318799 A318852 A052329 * A036624 A226907 A186952

Adjacent sequences: A199880 A199881 A199882 * A199884 A199885 A199886

Keyword

nonn,more

Author

Alois P. Heinz, Nov 11 2011

Extensions

a(22)-a(23) from Alois P. Heinz, Sep 26 2014

Status

approved