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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
1, 1, 4, 9, 156, 650, 5034, 26054, 4270304, 27617616, 198832320, 6251899104, 46466835072, 5033625978576, 37552294300416, 793996577407560, 6563364026374464, 13221301266369115200, 114481557932032050048, 1114510139284499182656, 109640692903857698897280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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), ... .

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

FORMULA

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

EXAMPLE

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

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

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.

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

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

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

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

CROSSREFS

Main diagonal of A215703.

Sequence in context: A061269 A061271 A084009 * A318616 A029738 A067072

Adjacent sequences:  A306736 A306737 A306738 * A306740 A306741 A306742

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 06 2019

STATUS

approved

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Last modified December 4 09:10 EST 2021. Contains 349484 sequences. (Running on oeis4.)