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A293473
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.
1
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
OFFSET
0,3
EXAMPLE
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].
MAPLE
# Function dx in A293472.
ListTools:-Flatten([seq(dx(2, n), n=0..8)]);
MATHEMATICA
(* Function dx in A293472. *)
Table[dx[2, n], {n, 0, 7}] // Flatten
CROSSREFS
T(n, 0) = A215524, T(n, n) = A000079.
More generally, consider the n-th derivative of x^(x^m).
A293472 (m=1), this seq. (m=2), A293474 (m=3).
Cf. A290268.
Sequence in context: A334016 A371241 A278259 * A353512 A257080 A099784
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 10 2017
STATUS
approved