A290268 Number of terms in the fully expanded n-th derivative of x^(x^2).
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
Keywords
Examples
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.
Programs
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Maple
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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Mathematica
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 *)
Formula
Conjectured g.f.: (1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + x^7 + 2*x^8 + x^9)/((1 - x)*(1 - x^2)*(1 - x^8)).
Conjecture: a(n) = (8*n^2 + 15*n + 14 + (n + 2)*(-1)^n + (2 - 4*sqrt(2)*sin(Pi*n/4))*sin(Pi*n/2))/16.
From Peter Luschny, Oct 09 2017: (Start) Assuming the conjecture:
a(n) = n^2/2 + n + 1 - (n mod 2)*(1/2 + floor((n + 1)/8)).
Signature of the linear recurrence: {0, 2, 0, -1, 0, 0, 0, 1, 0, -2, 0, 1}. (End)