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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.

%I A293472 #16 Oct 11 2017 10:24:09
%S A293472 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,
%T A293472 -42,378,210,280,105,42,7,1,944,-336,1512,560,560,168,56,8,1,-5112,
%U A293472 8496,-1512,4536,1260,1008,252,72,9,1
%N 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.
%e A293472 Triangle starts:
%e A293472 0: [  1]
%e A293472 1: [  1,   1]
%e A293472 2: [  2,   2,   1]
%e A293472 3: [  3,   6,   3,   1]
%e A293472 4: [  8,  12,  12,   4,   1]
%e A293472 5: [ 10,  40,  30,  20,   5,  1]
%e A293472 6: [ 54,  60, 120,  60,  30,  6, 1]
%e A293472 7: [-42, 378, 210, 280, 105, 42, 7, 1]
%e A293472 ...
%e A293472 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].
%p A293472 dx := proc(m, n) if n = 0 then return [1] fi;
%p A293472 subs(ln(x) = t, diff(x^(x^m), x$n)): subs(x = 1, %):
%p A293472 PolynomialTools:-CoefficientList(%,t) end:
%p A293472 ListTools:-Flatten([seq(dx(1, n), n=0..10)]);
%t A293472 dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
%t A293472 Table[dx[1, n], {n, 0, 7}] // Flatten
%Y A293472 More generally, consider the n-th derivative of x^(x^m). This is case m = 1.
%Y A293472 m      |  t = -1 |  t = 0  |  t = 1  | p(n, t) | related
%Y A293472 m = 1  | A203852 | A005727 | A293297 | A293472 | A293239
%Y A293472 m = 2  |    -    | A215524 |    -    | A293473 | A290268
%Y A293472 m = 3  |    -    | A215704 |    -    | A293474 |    -
%Y A293472 Cf. A215703.
%K A293472 sign,tabl
%O A293472 0,4
%A A293472 _Peter Luschny_, Oct 10 2017