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A295108 a(n) = (1/n) times the n-th derivative of the eighth tetration of x (power tower of order 8) x^^8 at x=1.

%I A295108 #12 Feb 16 2025 08:33:51
%S A295108 1,1,3,14,96,849,9362,123101,1847696,31252368,584145552,11981318986,
%T A295108 267050704368,6432872588700,166461202886456,4606491806670324,
%U A295108 135733988375074944,4243153626928512224,140252989224067186752,4887395830953148166784,179067423776388634331904
%N A295108 a(n) = (1/n) times the n-th derivative of the eighth tetration of x (power tower of order 8) x^^8 at x=1.
%H A295108 Alois P. Heinz, <a href="/A295108/b295108.txt">Table of n, a(n) for n = 1..412</a>
%H A295108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295108 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295108 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295108 a(n) = 1/n * [(d/dx)^n x^^8]_{x=1}.
%F A295108 a(n) = (n-1)! * [x^n] (x+1)^^8.
%F A295108 a(n) = 1/n * A277539(n).
%p A295108 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295108 a:= n-> (n-1)!*coeff(series(f(8), x, n+1), x, n):
%p A295108 seq(a(n), n=1..23);
%t A295108 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295108 a[n_] := (n - 1)!*SeriesCoefficient[f[8], {x, 0, n}];
%t A295108 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295108 Column k=8 of A295028.
%Y A295108 Cf. A277539.
%K A295108 nonn
%O A295108 1,3
%A A295108 _Alois P. Heinz_, Nov 14 2017