cp's OEIS Frontend

A295109 a(n) = (1/n) times the n-th derivative of the ninth tetration of x (power tower of order 9) x^^9 at x=1.

%I A295109 #10 Feb 16 2025 08:33:51
%S A295109 1,1,3,14,96,849,9362,123101,1888016,32703888,631924752,13408344586,
%T A295109 310042428528,7748365327260,208162961545016,5980417481391924,
%U A295109 182983003358805504,5940081852766157024,203920022890052114112,7381110402800795329344,280947854368982073172224
%N A295109 a(n) = (1/n) times the n-th derivative of the ninth tetration of x (power tower of order 9) x^^9 at x=1.
%H A295109 Alois P. Heinz, <a href="/A295109/b295109.txt">Table of n, a(n) for n = 1..409</a>
%H A295109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295109 a(n) = 1/n * [(d/dx)^n x^^9]_{x=1}.
%F A295109 a(n) = (n-1)! * [x^n] (x+1)^^9.
%F A295109 a(n) = 1/n * A277540(n).
%p A295109 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295109 a:= n-> (n-1)!*coeff(series(f(9), x, n+1), x, n):
%p A295109 seq(a(n), n=1..23);
%t A295109 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295109 a[n_] := (n - 1)!*SeriesCoefficient[f[9], {x, 0, n}];
%t A295109 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295109 Column k=9 of A295028.
%Y A295109 Cf. A277540.
%K A295109 nonn
%O A295109 1,3
%A A295109 _Alois P. Heinz_, Nov 14 2017