OFFSET
0,4
EXAMPLE
Triangle starts:
0: [ 1]
1: [ 1, 1]
2: [ 2, 2, 1]
3: [ 3, 6, 3, 1]
4: [ 8, 12, 12, 4, 1]
5: [ 10, 40, 30, 20, 5, 1]
6: [ 54, 60, 120, 60, 30, 6, 1]
7: [-42, 378, 210, 280, 105, 42, 7, 1]
...
For n = 3, the 3rd derivative of x^x is p(3,x,t) = x^x*t^3 + 3*x^x*t^2 + 3*x^x*t + x^x + 3*x^x*t/x + 3*x^x/x - x^x/x^2 where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 3 + 6*t + 3*t^2 + t^3 with coefficients [3, 6, 3, 1].
MAPLE
dx := proc(m, n) if n = 0 then return [1] fi;
subs(ln(x) = t, diff(x^(x^m), x$n)): subs(x = 1, %):
PolynomialTools:-CoefficientList(%, t) end:
ListTools:-Flatten([seq(dx(1, n), n=0..10)]);
MATHEMATICA
dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
Table[dx[1, n], {n, 0, 7}] // Flatten
CROSSREFS
More generally, consider the n-th derivative of x^(x^m). This is case m = 1.
m | t = -1 | t = 0 | t = 1 | p(n, t) | related
Cf. A215703.
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 10 2017
STATUS
approved