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A295106 a(n) = (1/n) times the n-th derivative of the sixth tetration of x (power tower of order 6) x^^6 at x=1.

%I A295106 #10 Feb 16 2025 08:33:51
%S A295106 1,1,3,14,96,849,8642,102941,1373936,20607888,340516992,6173590906,
%T A295106 121502258688,2583247609500,58940269686776,1437019737587004,
%U A295106 37267502536335744,1024420897710717344,29745405670928179392,909702365350759274304,29224500667382460549504
%N A295106 a(n) = (1/n) times the n-th derivative of the sixth tetration of x (power tower of order 6) x^^6 at x=1.
%H A295106 Alois P. Heinz, <a href="/A295106/b295106.txt">Table of n, a(n) for n = 1..423</a>
%H A295106 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295106 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295106 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295106 a(n) = 1/n * [(d/dx)^n x^^6]_{x=1}.
%F A295106 a(n) = (n-1)! * [x^n] (x+1)^^6.
%F A295106 a(n) = 1/n * A211205(n).
%p A295106 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295106 a:= n-> (n-1)!*coeff(series(f(6), x, n+1), x, n):
%p A295106 seq(a(n), n=1..23);
%t A295106 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295106 a[n_] := (n - 1)!*SeriesCoefficient[f[6], {x, 0, n}];
%t A295106 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295106 Column k=6 of A295028.
%Y A295106 Cf. A211205.
%K A295106 nonn
%O A295106 1,3
%A A295106 _Alois P. Heinz_, Nov 14 2017