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.

1, 1, 3, 14, 96, 849, 9362, 123101, 1888016, 32703888, 631924752, 13408344586, 310042428528, 7748365327260, 208162961545016, 5980417481391924, 182983003358805504, 5940081852766157024, 203920022890052114112, 7381110402800795329344, 280947854368982073172224

Offset: 1

Author

Alois P. Heinz, Nov 14 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..409
• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Knuth's up-arrow notation
• Wikipedia, Tetration

Crossrefs

Column k=9 of A295028.

Cf. A277540.

Programs

Maple

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(9), 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[9], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) = 1/n * [(d/dx)^n x^^9]_{x=1}.

a(n) = (n-1)! * [x^n] (x+1)^^9.

a(n) = 1/n * A277540(n).