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A295104 a(n) = (1/n) times the n-th derivative of the fourth tetration of x (power tower of order 4) x^^4 at x=1.

%I A295104 #10 Feb 16 2025 08:33:51
%S A295104 1,1,3,14,72,489,3722,33641,334520,3761688,45898272,615641806,
%T A295104 8863726704,137786878644,2279658872696,40229212948404,750433323448128,
%U A295104 14801457167223872,306869893647304896,6683254543551623904,152281219079726183040,3626445842114839589952
%N A295104 a(n) = (1/n) times the n-th derivative of the fourth tetration of x (power tower of order 4) x^^4 at x=1.
%C A295104 First term < 0: a(329).
%H A295104 Alois P. Heinz, <a href="/A295104/b295104.txt">Table of n, a(n) for n = 1..452</a>
%H A295104 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295104 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295104 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295104 a(n) = 1/n * [(d/dx)^n x^^4]_{x=1}.
%F A295104 a(n) = (n-1)! * [x^n] (x+1)^^4.
%F A295104 a(n) = 1/n * A179405(n).
%p A295104 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295104 a:= n-> (n-1)!*coeff(series(f(4), x, n+1), x, n):
%p A295104 seq(a(n), n=1..23);
%t A295104 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295104 a[n_] := (n - 1)!*SeriesCoefficient[f[4], {x, 0, n}];
%t A295104 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295104 Column k=4 of A295028.
%Y A295104 Cf. A179405.
%K A295104 sign
%O A295104 1,3
%A A295104 _Alois P. Heinz_, Nov 14 2017