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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.

%I A306739 #14 Sep 15 2022 05:13:12
%S A306739 1,1,4,9,156,650,5034,26054,4270304,27617616,198832320,6251899104,
%T A306739 46466835072,5033625978576,37552294300416,793996577407560,
%U A306739 6563364026374464,13221301266369115200,114481557932032050048,1114510139284499182656,109640692903857698897280
%N 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.
%C A306739 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 A306739 Alois P. Heinz, <a href="/A306739/b306739.txt">Table of n, a(n) for n = 0..400</a>
%F A306739 a(n) = A215703(n,n+1).
%e A306739 a(0) = x_{x=1} = 1.
%e A306739 a(1) = (d/dx x^x)_{x=1} = (x^x*(log(x)+1))_{x=1} = log(1)+1 = 1.
%e A306739 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 A306739 a(3) = (d^3/dx^3 x^(x^x))_{x=1} = 9.
%e A306739 a(4) = (d^4/dx^4 x^(x^3))_{x=1} = 156.
%p A306739 T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
%p A306739 g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
%p A306739       seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
%p A306739       combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
%p A306739     end:
%p A306739 f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
%p A306739       nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
%p A306739     end():
%p A306739 a:= n-> n!*coeff(series(subs(x=x+1, f(n+1)), x, n+1), x, n):
%p A306739 seq(a(n), n=0..23);
%t A306739 T[n_] := If[n == 1, {x}, Map[x^# &, g[n - 1, n - 1]]];
%t A306739 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}]];
%t A306739 f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
%t A306739 a[n_] := n!*SeriesCoefficient[f[n+1] /. x -> x+1, {x, 0, n}];
%t A306739 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Sep 15 2022, after _Alois P. Heinz_ *)
%Y A306739 Main diagonal of A215703.
%K A306739 nonn
%O A306739 0,3
%A A306739 _Alois P. Heinz_, Mar 06 2019