A295105 - OEIS

%I #12 Feb 16 2025 08:33:51

%S 1,1,3,14,96,729,6842,71861,869936,11639208,172823592,2797075786,

%T 49197117648,931938081060,18931744650296,410210251648404,

%U 9443418462484224,230108297407058144,5915598580747789632,159987606074180375904,4539874767394840811904

%N a(n) = (1/n) times the n-th derivative of the fifth tetration of x (power tower of order 5) x^^5 at x=1.

%H Alois P. Heinz, <a href="/A295105/b295105.txt">Table of n, a(n) for n = 1..434</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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 a(n) = 1/n * [(d/dx)^n x^^5]_{x=1}.

%F a(n) = (n-1)! * [x^n] (x+1)^^5.

%F a(n) = 1/n * A179505(n).

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

%p a:= n-> (n-1)!*coeff(series(f(5), x, n+1), x, n):

%p seq(a(n), n=1..23);

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

%t a[n_] := (n - 1)!*SeriesCoefficient[f[5], {x, 0, n}];

%t Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)

%Y Column k=5 of A295028.

%Y Cf. A179505.

%K nonn,changed

%O 1,3

%A _Alois P. Heinz_, Nov 14 2017