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A295103 a(n) = (1/n) times the n-th derivative of the third tetration of x (power tower of order 3) x^^3 at x=1.

%I A295103 #15 Feb 16 2025 08:33:51
%S A295103 1,1,3,8,36,159,932,5627,40016,302364,2510712,22623490,213486864,
%T A295103 2227719948,23388469400,277570328040,3182959484736,42530335589088,
%U A295103 523078873327872,7846745537655360,101370634558327680,1717052148685665792,22657314273376353408
%N A295103 a(n) = (1/n) times the n-th derivative of the third tetration of x (power tower of order 3) x^^3 at x=1.
%C A295103 First term < 0: a(33) = -26329560314038014690778779463680.
%H A295103 Alois P. Heinz, <a href="/A295103/b295103.txt">Table of n, a(n) for n = 1..453</a>
%H A295103 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295103 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295103 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295103 a(n) = 1/n * [(d/dx)^n x^^3]_{x=1}.
%F A295103 a(n) = (n-1)! * [x^n] (x+1)^^3.
%F A295103 a(n) = 1/n * A179230(n).
%p A295103 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295103 a:= n-> (n-1)!*coeff(series(f(3), x, n+1), x, n):
%p A295103 seq(a(n), n=1..23);
%t A295103 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295103 a[n_] := (n - 1)!*SeriesCoefficient[f[3], {x, 0, n}];
%t A295103 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295103 Column k=3 of A295028.
%Y A295103 Cf. A179230.
%K A295103 sign
%O A295103 1,3
%A A295103 _Alois P. Heinz_, Nov 14 2017