A005168 n-th derivative of x^x at 1, divided by n.

1, 1, 1, 2, 2, 9, -6, 118, -568, 4716, -38160, 358126, -3662088, 41073096, -500013528, 6573808200, -92840971200, 1402148010528, -22554146644416, 385014881294496, -6952611764874240, 132427188835260480, -2653529921603890560, 55802195178451990896

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• sign
• easy
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..400 (first 100 terms from T. D. Noe)
• R. K. Guy, Letter to N. J. A. Sloane, 1986
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Crossrefs

Cf. A005727.

Column k=2 of A295027 (for n>1), A295028.

Programs

Maple

a:= n-> (n-1)! *coeftayl(x^x, x=1, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 18 2012

Mathematica

Rest[(NestList[ Factor[ D[ #1, x]] &, x^x, 23] /. (x -> 1))/Range[0, 23]] (* Robert G. Wilson v, Aug 10 2010 *)

Python

from sympy import var, diff
x = var('x')
y = x**x
l = [[y:=diff(y),y.subs(x,1)/(n+1)][1] for n in range(10)]
print(l) # Nicholas Stefan Georgescu, Mar 02 2023

Extensions

One more term from Robert G. Wilson v, Aug 10 2010