A295103 a(n) = (1/n) times the n-th derivative of the third tetration of x (power tower of order 3) x^^3 at x=1.

1, 1, 3, 8, 36, 159, 932, 5627, 40016, 302364, 2510712, 22623490, 213486864, 2227719948, 23388469400, 277570328040, 3182959484736, 42530335589088, 523078873327872, 7846745537655360, 101370634558327680, 1717052148685665792, 22657314273376353408

Offset: 1

Author

Alois P. Heinz, Nov 14 2017

Keywords

• sign

Comments

First term < 0: a(33) = -26329560314038014690778779463680.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..453
• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Knuth's up-arrow notation
• Wikipedia, Tetration

Crossrefs

Column k=3 of A295028.

Cf. A179230.

Programs

Maple

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(3), x, n+1), x, n):
seq(a(n), n=1..23);

Mathematica

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[3], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) = 1/n * [(d/dx)^n x^^3]_{x=1}.

a(n) = (n-1)! * [x^n] (x+1)^^3.

a(n) = 1/n * A179230(n).