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A277539 n-th derivative of the eighth tetration of x (power tower of order 8) x^^8 at x=1.

%I A277539 #15 Feb 16 2025 08:33:37
%S A277539 1,1,2,9,56,480,5094,65534,984808,16629264,312523680,6425601072,
%T A277539 143775827832,3471659156784,90060216241800,2496918043296840,
%U A277539 73703868906725184,2307477802376274048,76376765284713220032,2664806795257276548288,97747916619062963335680
%N A277539 n-th derivative of the eighth tetration of x (power tower of order 8) x^^8 at x=1.
%H A277539 Alois P. Heinz, <a href="/A277539/b277539.txt">Table of n, a(n) for n = 0..400</a>
%H A277539 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A277539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A277539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A277539 E.g.f.: (x+1)^^8.
%p A277539 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A277539 a:= n-> n!*coeff(series(f(8), x, n+1), x, n):
%p A277539 seq(a(n), n=0..25);
%t A277539 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A277539 a[n_] := n!*SeriesCoefficient[f[8], {x, 0, n}];
%t A277539 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A277539 Column k=8 of A277537.
%Y A277539 Cf. A215703, A295108.
%K A277539 nonn
%O A277539 0,3
%A A277539 _Alois P. Heinz_, Oct 19 2016