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A293474 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^(x^3), evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.

%I A293474 #15 Oct 11 2017 10:24:05
%S A293474 1,1,3,6,12,9,27,78,81,27,156,564,720,432,81,1110,4320,6930,5400,2025,
%T A293474 243,8322,37260,68940,66420,34830,8748,729,70098,347382,722610,824040,
%U A293474 541485,200718,35721,2187
%N A293474 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^(x^3), evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.
%e A293474 Triangle start:
%e A293474 0: [    1]
%e A293474 1: [    1,      3]
%e A293474 2: [    6,     12,      9]
%e A293474 3: [   27,     78,     81,     27]
%e A293474 4: [  156,    564,    720,    432,     81]
%e A293474 5: [ 1110,   4320,   6930,   5400,   2025,    243]
%e A293474 6: [ 8322,  37260,  68940,  66420,  34830,   8748,   729]
%e A293474 7: [70098, 347382, 722610, 824040, 541485, 200718, 35721, 2187]
%e A293474 ...
%e A293474 For n = 3, the 3rd derivative of x^(x^3) is p(3,x,t) = 27*t^3*x^6*x^(x^3) + 27*t^2*x^6*x^(x^3) + 9*t*x^6*x^(x^3) + x^6*x^(x^3) + 54*t^2*x^3*x^(x^3) + 63*t*x^3*x^(x^3) + 15*x^3*x^(x^3) + 6*t*x^(x^3) + 11*x^(x^3) where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 27 + 78*t + 81*t^2 + 27*t^3 with coefficients [27, 78, 81, 27].
%p A293474 # Function dx in A293472.
%p A293474 ListTools:-Flatten([seq(dx(3, n), n=0..8)]);
%t A293474 (* Function dx in A293472. *)
%t A293474 Table[dx[3, n], {n, 0, 7}] // Flatten
%Y A293474 T(n, 0) = A215704, T(n, n) = A000244.
%Y A293474 More generally, consider the n-th derivative of x^(x^m).
%Y A293474 A293472 (m=1), A293472 (m=2), this seq. (m=3).
%K A293474 sign,tabl
%O A293474 0,3
%A A293474 _Peter Luschny_, Oct 10 2017