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.
Original entry on oeis.org
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, -42, 378, 210, 280, 105, 42, 7, 1, 944, -336, 1512, 560, 560, 168, 56, 8, 1, -5112, 8496, -1512, 4536, 1260, 1008, 252, 72, 9, 1
Offset: 0
Triangle starts:
0: [ 1]
1: [ 1, 1]
2: [ 2, 2, 1]
3: [ 3, 6, 3, 1]
4: [ 8, 12, 12, 4, 1]
5: [ 10, 40, 30, 20, 5, 1]
6: [ 54, 60, 120, 60, 30, 6, 1]
7: [-42, 378, 210, 280, 105, 42, 7, 1]
...
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].
More generally, consider the n-th derivative of x^(x^m). This is case m = 1.
m | t = -1 | t = 0 | t = 1 | p(n, t) | related
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dx := proc(m, n) if n = 0 then return [1] fi;
subs(ln(x) = t, diff(x^(x^m), x$n)): subs(x = 1, %):
PolynomialTools:-CoefficientList(%,t) end:
ListTools:-Flatten([seq(dx(1, n), n=0..10)]);
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dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
Table[dx[1, n], {n, 0, 7}] // Flatten
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.
Original entry on oeis.org
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
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].
More generally, consider the n-th derivative of x^(x^m).
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# Function dx in A293472.
ListTools:-Flatten([seq(dx(2, n), n=0..8)]);
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(* Function dx in A293472. *)
Table[dx[2, n], {n, 0, 7}] // Flatten
Showing 1-2 of 2 results.