A262400 - OEIS

A262400

Let f(x) = 1 + Sum_{j>=4} (|mu(j)| - |mu(j-1)|)*x^j, where mu(n) is the Möbius function (A008683). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

%I #32 Dec 04 2015 23:51:03

%S 0,0,0,0,-4,5,0,0,-12,9,5,0,-28,39,0,-10,-60,102,-45,0,-119,252,-132,

%T 0,-252,580,-403,9,-424,1363,-1210,248,-828,3003,-3332,1195,-1729,

%U 6697,-8740,4290,-3807,14514,-22176,13889,-9288,31049,-54142,41501,-25260,66885,-129570

%N Let f(x) = 1 + Sum_{j>=4} (|mu(j)| - |mu(j-1)|)*x^j, where mu(n) is the Möbius function (A008683). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

%C Function f(x) is connected with the density h of the exponentially squarefree numbers (A209061). Specifically, for h = Product_{prime p} f(1/p), this sequence allows the calculation of h with very high accuracy (cf. A262276).

%H Juan Arias-de-Reyna, <a href="/A262400/b262400.txt">Table of n, a(n) for n = 0..3000</a>

%t M = 50; (* to get the first 51 terms *)

%t f = 1 + Sum[(MoebiusMu[n]^2 - MoebiusMu[n - 1]^2) x^n, {n, 4, M}];

%t S = Series[Log[f], {x, 0, M}];

%t A262400[nn_] := CoefficientList[S, x][[nn + 1]] nn;

%t Table[A262400[n], {n, 0, M}]

%Y Cf. A008683, A209061, A262276.

%K sign

%O 0,5

%A _Juan Arias-de-Reyna_, Sep 21 2015