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.

1, 1, 4, 9, 156, 650, 5034, 26054, 4270304, 27617616, 198832320, 6251899104, 46466835072, 5033625978576, 37552294300416, 793996577407560, 6563364026374464, 13221301266369115200, 114481557932032050048, 1114510139284499182656, 109640692903857698897280

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);

Mathematica

T[n_] := If[n == 1, {x}, Map[x^# &, g[n - 1, n - 1]]];
g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten@ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[Range[Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
a[n_] := n!*SeriesCoefficient[f[n+1] /. x -> x+1, {x, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 15 2022, after Alois P. Heinz *)

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