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 A306739 n-th derivative of f_{n+1} at x=1, where f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways. 2

%I

%S 1,1,4,9,156,650,5034,26054,4270304,27617616,198832320,6251899104,

%T 46466835072,5033625978576,37552294300416,793996577407560,

%U 6563364026374464,13221301266369115200,114481557932032050048,1114510139284499182656,109640692903857698897280

%N n-th derivative of f_{n+1} at x=1, where f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.

%C The ordering of the functions f_k is defined in A215703: f_1, f_2, ... = x, x^x, x^(x^2), x^(x^x), x^(x^3), x^(x^x*x), x^(x^(x^2)), x^(x^(x^x)), x^(x^4), x^(x^x*x^2), ... .

%H Alois P. Heinz, <a href="/A306739/b306739.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) = A215703(n,n+1).

%e a(0) = x_{x=1} = 1.

%e a(1) = (d/dx x^x)_{x=1} = (x^x*(log(x)+1))_{x=1} = log(1)+1 = 1.

%e a(2) = (d^2/dx^2 x^(x^2))_{x=1} = (x^(x^2) * (2*x*log(x)+x)^2 + x^(x^2) * (2*log(x)+3))_{x=1} = (2*log(1)+1)^2 + 2*log(1)+3 = 4.

%e a(3) = (d^3/dx^3 x^(x^x))_{x=1} = 9.

%e a(4) = (d^4/dx^4 x^(x^3))_{x=1} = 156.

%p T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1\$2))) end:

%p g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(

%p seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=

%p combinat[choose]([\$1..nops(T(i))+j-1], j)), j=0..n/i)])

%p end:

%p f:= proc() local i, l; i, l:= 0, []; proc(n) while n>

%p nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end

%p end():

%p a:= n-> n!*coeff(series(subs(x=x+1, f(n+1)), x, n+1), x, n):

%p seq(a(n), n=0..23);

%Y Main diagonal of A215703.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 06 2019

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Last modified January 20 21:15 EST 2022. Contains 350472 sequences. (Running on oeis4.)