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$n)): subs(x = 1, %):
PolynomialTools:-CoefficientList(%,t) end:
ListTools:-Flatten([seq(dx(1, n), n=0..10)]);
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dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
Table[dx[1, n], {n, 0, 7}] // Flatten
A281434
Number of terms in the fully expanded n-th derivative of x^(x^x).
Original entry on oeis.org
1, 3, 12, 30, 64, 113, 188, 285, 415, 577, 780, 1017, 1312, 1648, 2044, 2489, 3008, 3583, 4236, 4953, 5760, 6638, 7611, 8664, 9822, 11069, 12426, 13880, 15455, 17131, 18940, 20855, 22912, 25083, 27404, 29844, 32448, 35178, 38075, 41109, 44320, 47672, 51212
Offset: 0
For n=1, the 1st derivative of x^(x^x) is x^(x^x+x-1) + x^(x^x+x)*log(x) + x^(x^x+x)*log^2(x), so a(1) = 3.
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Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^x, 42]]]
A290268
Number of terms in the fully expanded n-th derivative of x^(x^2).
Original entry on oeis.org
1, 2, 5, 8, 13, 18, 25, 31, 41, 49, 61, 71, 85, 97, 113, 126, 145, 160, 181, 198, 221, 240, 265, 285, 313, 335, 365, 389, 421, 447, 481, 508, 545, 574, 613, 644, 685, 718, 761, 795, 841, 877, 925, 963, 1013, 1053, 1105, 1146, 1201, 1244, 1301, 1346, 1405, 1452
Offset: 0
For n = 2, the 2nd derivative of x^(x^2) is 3*x^(x^2) + 2*x^(x^2)*log(x) + x^(x^2+2) + 4*x^(x^2+2)*log(x) + 4*x^(x^2+2)*log^2(x), so a(2) = 5.
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a := n -> `if`(n=0, 1, nops(expand(diff(x^(x^2), x$n)))):
seq(a(n), n = 0..30); # Peter Luschny, Oct 08 2017
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Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^2, 53]]]
(* Use it only to check the conjecture, not to compute the values: *)
LinearRecurrence[{0,2,0,-1,0,0,0,1,0,-2,0,1}, {1,2,5,8,13,18,25,31,41,49,61,71}, 54] (* Peter Luschny, Oct 09 2017 *)
Showing 1-3 of 3 results.