A293472 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^x, evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 8, 12, 12, 4, 1, 10, 40, 30, 20, 5, 1, 54, 60, 120, 60, 30, 6, 1, -42, 378, 210, 280, 105, 42, 7, 1, 944, -336, 1512, 560, 560, 168, 56, 8, 1, -5112, 8496, -1512, 4536, 1260, 1008, 252, 72, 9, 1

Offset: 0

Author

Peter Luschny, Oct 10 2017

Keywords

• sign
• tabl

Examples

			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].
		

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

m = 1 | A203852 | A005727 | A293297 | A293472 | A293239

m = 2 | - | A215524 | - | A293473 | A290268

m = 3 | - | A215704 | - | A293474 | -

Cf. A215703.

Programs

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