A295104 a(n) = (1/n) times the n-th derivative of the fourth tetration of x (power tower of order 4) x^^4 at x=1.

1, 1, 3, 14, 72, 489, 3722, 33641, 334520, 3761688, 45898272, 615641806, 8863726704, 137786878644, 2279658872696, 40229212948404, 750433323448128, 14801457167223872, 306869893647304896, 6683254543551623904, 152281219079726183040, 3626445842114839589952

Offset: 1

Author

Alois P. Heinz, Nov 14 2017

Keywords

• sign

Comments

First term < 0: a(329).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..452
• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Knuth's up-arrow notation
• Wikipedia, Tetration

Crossrefs

Column k=4 of A295028.

Cf. A179405.

Programs

Maple

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(4), 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[4], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) = 1/n * [(d/dx)^n x^^4]_{x=1}.

a(n) = (n-1)! * [x^n] (x+1)^^4.

a(n) = 1/n * A179405(n).