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A277540 n-th derivative of the ninth tetration of x (power tower of order 9) x^^9 at x=1.

%I A277540 #15 Feb 16 2025 08:33:37
%S A277540 1,1,2,9,56,480,5094,65534,984808,16992144,327038880,6951172272,
%T A277540 160900135032,4030551570864,108477114581640,3122444423175240,
%U A277540 95686679702270784,3110711057099693568,106921473349790826432,3874480434910990168128,147622208056015906586880
%N A277540 n-th derivative of the ninth tetration of x (power tower of order 9) x^^9 at x=1.
%H A277540 Alois P. Heinz, <a href="/A277540/b277540.txt">Table of n, a(n) for n = 0..400</a>
%H A277540 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A277540 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A277540 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A277540 E.g.f.: (x+1)^^9.
%p A277540 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A277540 a:= n-> n!*coeff(series(f(9), x, n+1), x, n):
%p A277540 seq(a(n), n=0..25);
%t A277540 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A277540 a[n_] := n!*SeriesCoefficient[f[9], {x, 0, n}];
%t A277540 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)
%Y A277540 Column k=9 of A277537.
%Y A277540 Cf. A215703, A295109.
%K A277540 nonn
%O A277540 0,3
%A A277540 _Alois P. Heinz_, Oct 19 2016