%I #14 May 30 2018 03:29:32

%S 1,1,2,9,56,480,5094,65534,984808,16992144,327038880,6951172272,

%T 160900135032,4030551570864,108477114581640,3122444423175240,

%U 95686679702270784,3110711057099693568,106921473349790826432,3874480434910990168128,147622208056015906586880

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

%H Alois P. Heinz, <a href="/A277540/b277540.txt">Table of n, a(n) for n = 0..400</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>

%F E.g.f.: (x+1)^^9.

%p f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:

%p a:= n-> n!*coeff(series(f(9), x, n+1), x, n):

%p seq(a(n), n=0..25);

%t f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];

%t a[n_] := n!*SeriesCoefficient[f[9], {x, 0, n}];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 30 2018, from Maple *)

%Y Column k=9 of A277537.

%Y Cf. A215703, A295109.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 19 2016