OFFSET
0,2
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)
EXAMPLE
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.
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 06 2017
STATUS
approved