A215707
n-th derivative of ((x^x)^x)^(x^x) at x=1.
Original entry on oeis.org
1, 1, 6, 33, 228, 1880, 17742, 187124, 2176360, 27617616, 378764280, 5574170712, 87491513304, 1457433784560, 25654258467432, 475431102931080, 9246150139382400, 188172595998890688, 3997389233216787264, 88440294467474068608, 2033755519425292281600
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, x^(x^x*x^2) ), x, n+1), x, n):
seq(a(n), n=0..25);
A215708
n-th derivative of (x^(x^x))^(x^x) at x=1.
Original entry on oeis.org
1, 1, 4, 24, 148, 1180, 10428, 106876, 1198160, 14843808, 198832320, 2877693984, 44545268832, 734929736736, 12852051257472, 237372559264320, 4614124211454720, 94103610003019008, 2008507968212696064, 44748953208031094784, 1038646472528272158720
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, x^((x^x)^2) ), x, n+1), x, n):
seq(a(n), n=0..25);
A215709
n-th derivative of (x^x)^((x^x)^x) at x=1.
Original entry on oeis.org
1, 1, 4, 24, 172, 1420, 13968, 154336, 1914288, 26108208, 388596960, 6251899104, 108088087776, 1995840455232, 39183950494752, 814399382073120, 17856182764554240, 411671923447488768, 9952212794293198080, 251646630845685827328, 6640389412581544588800
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^2)*x) ), x, n+1), x, n):
seq(a(n), n=0..25);
A215710
n-th derivative of x^(((x^x)^x)^x) at x=1.
Original entry on oeis.org
1, 1, 2, 21, 152, 1360, 15174, 184296, 2538584, 39097296, 656793720, 12021152616, 237610299288, 5033625978576, 113810068532328, 2733480292962600, 69463846973884800, 1861656629684769600, 52458209090931835584, 1549997983761108724224, 47908467697220966937600
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^3)) ), x, n+1), x, n):
seq(a(n), n=0..25);
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
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