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A293473
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Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^(x^2), evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.
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1
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1, 1, 2, 4, 6, 4, 12, 30, 24, 8, 52, 144, 156, 80, 16, 240, 760, 1020, 680, 240, 32, 1188, 4440, 6720, 5640, 2640, 672, 64, 6804, 26712, 47040, 45640, 26880, 9408, 1792, 128, 38960, 175392, 338016, 376320, 261520, 115584, 31360, 4608, 256
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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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 starts:
0: [ 1]
1: [ 1, 2]
2: [ 4, 6, 4]
3: [ 12, 30, 24, 8]
4: [ 52, 144, 156, 80, 16]
5: [ 240, 760, 1020, 680, 240, 32]
6: [1188, 4440, 6720, 5640, 2640, 672, 64]
7: [6804, 26712, 47040, 45640, 26880, 9408, 1792, 128]
...
For n = 3, the 3rd derivative of x^(x^2) is p(3,x,t) = 8*t^3*x^3*x^(x^2) + 12*t^2*x^3*x^(x^2) + 6*t*x^3*x^(x^2) + 12*t^2*x*x^(x^2) + x^3*x^(x^2) + 24*t*x*x^(x^2) + 9*x*x^(x^2) + 2*x^(x^2)/x where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 12 + 30*t + 24*t^2 + 8*t^3 with coefficients [12, 30, 24, 8].
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MAPLE
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ListTools:-Flatten([seq(dx(2, n), n=0..8)]);
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MATHEMATICA
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Table[dx[2, 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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