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A293474
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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.
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2
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1, 1, 3, 6, 12, 9, 27, 78, 81, 27, 156, 564, 720, 432, 81, 1110, 4320, 6930, 5400, 2025, 243, 8322, 37260, 68940, 66420, 34830, 8748, 729, 70098, 347382, 722610, 824040, 541485, 200718, 35721, 2187
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table;
graph;
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history;
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OFFSET
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0,3
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LINKS
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EXAMPLE
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Triangle start:
0: [ 1]
1: [ 1, 3]
2: [ 6, 12, 9]
3: [ 27, 78, 81, 27]
4: [ 156, 564, 720, 432, 81]
5: [ 1110, 4320, 6930, 5400, 2025, 243]
6: [ 8322, 37260, 68940, 66420, 34830, 8748, 729]
7: [70098, 347382, 722610, 824040, 541485, 200718, 35721, 2187]
...
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].
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MAPLE
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ListTools:-Flatten([seq(dx(3, n), n=0..8)]);
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MATHEMATICA
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Table[dx[3, n], {n, 0, 7}] // Flatten
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CROSSREFS
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More generally, consider the n-th derivative of x^(x^m).
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KEYWORD
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AUTHOR
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STATUS
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approved
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