A215643
n-th derivative of x^((x^(x^x))^x) at x=1.
Original entry on oeis.org
1, 1, 2, 15, 104, 890, 8814, 100660, 1288048, 18337680, 286674960, 4882660464, 89880715704, 1777384045944, 37552294300416, 843830334815640, 20086549955304384, 504750167170162944, 13348550475903813120, 370499740676381737728, 10766442934111876381440
Offset: 0
-
a:= n-> n!*coeff(series(subs(x=x+1, x^((x^(x^x))^x) ), x, n+1), x, n):
seq(a(n), n=0..30);
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m = 20;
CoefficientList[(x+1)^(((x+1)^((x+1)^(x+1)))^(x+1)) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)
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
A215629
n-th derivative of x^(x^((x^x)^x)) at x=1.
Original entry on oeis.org
1, 1, 2, 9, 80, 660, 6714, 77084, 1005640, 14572944, 233086920, 4066783512, 76906345944, 1566049091568, 34153725715368, 793996577407560, 19595885746343808, 511550462381982528, 14080034085212120256, 407434430977558009344, 12363449947108075756800
Offset: 0
-
a:= n-> n!*coeff(series(subs(x=x+1, x^(x^((x^x)^x)) ), x, n+1), x, n):
seq(a(n), n=0..30);
-
m = 20;
CoefficientList[(x+1)^((x+1)^(((x+1)^(x+1))^(x+1))) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)
A215691
n-th derivative of (x^x)^(x^(x^x)) at x=1.
Original entry on oeis.org
1, 1, 4, 18, 124, 950, 8688, 89600, 1038392, 13309272, 186471480, 2837173152, 46466835072, 815532508440, 15246845864040, 302533865599800, 6344720827608384, 140208886623418752, 3254819745378435264, 79172189409906466560, 2013138139856523598080
Offset: 0
-
a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^(x^(x^x))), x, n+1), x, n):
seq(a(n), n=0..30);
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m = 20;
CoefficientList[(x+1)^((x+1)^((x+1)^(x+1)+1)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)
A215705
n-th derivative of x^((x^x)^x) at x=1.
Original entry on oeis.org
1, 1, 2, 15, 80, 590, 5034, 47110, 511216, 6019416, 77899320, 1092871824, 16459538952, 265695302808, 4560878625744, 83020743848760, 1595943389477760, 32291354360340672, 685838983512807360, 15248888357184824256, 354130117874225585280, 8571971677758345319680
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, x^((x^x)^x) ), x, n+1), x, n):
seq(a(n), n=0..25);
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With[{nn=30},CoefficientList[Series[(x+1)^((x+1)^((x+1)^2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 30 2013 *)
A215706
n-th derivative of (((x^x)^x)^x)^x at x=1.
Original entry on oeis.org
1, 1, 8, 48, 344, 3160, 31776, 349440, 4270304, 56343456, 794577600, 11975388480, 191431339392, 3225851451264, 57152333898240, 1061030230525440, 20569247105571840, 415385999498849280, 8719401647417757696, 189836589049809334272, 4278839631584572661760
Offset: 0
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a:= n-> n!*coeff(series(subs(x=x+1, (((x^x)^x)^x)^x ), x, n+1), x, n):
seq(a(n), n=0..25);
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
-
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
-
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
-
a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^3)) ), x, n+1), x, n):
seq(a(n), n=0..25);
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