A271726 Let f(x) = 1 -x^3+ Sum_{j>=2} (x^(2^j)-x^(1+2^j)). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

0, 0, -3, 4, -5, -3, 7, -4, -3, 5, -11, 1, 13, -21, 7, 28, -51, 33, 19, -91, 109, -33, -115, 209, -155, -65, 321, -381, 87, 407, -713, 476, 349, -1207, 1227, -35, -1739, 2603, -1277, -1979, 4797, -4161, -903, 7451, -9713, 3427, 9165, -18575, 14021, 6455, -29991, 34779

Offset: 1

Author

Juan Arias-de-Reyna, Apr 13 2016

Keywords

• sign

Comments

Function f(x) is connected with the density h of exponentially (2^n)-numbers (A138302). Specifically, for h = Product_{prime p} f(1/p), this sequence allows the calculation of h with very high accuracy (cf. A271727).

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..5000
• Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Programs

Mathematica

M = 6; K = 50; (* To get the first  50 terms *)
f = 1 - x^3 + Sum[x^(2^r) - x^(2^r + 1), {r, 2, M}];
S = Series[Log[f], {x, 0, K}];
If[2^M <= K, Print["Warning: 2^M should be greater than K and it is not. Change parameters."]];
L = CoefficientList[S, x];
A271726[n_] := n L[[n + 1]];
Table[A271726[n], {n, 1, K}]