OFFSET
1,4
COMMENTS
Conjecture: All coefficients are positive and strictly increasing starting from 5th term.
Probably the k-values for which c(i) calculated from the equation Product_{i>=1}(1-c(i)*x^i) = Sum_{i>=0}(möbius(i+k)*x^i) is positive and increasing are [1, 2, 27, 28, 39, 40, 41, 58, 65, 69, ...].
LINKS
Robert Israel, Table of n, a(n) for n = 1..1000
MAPLE
with(ListTools): L := product(1-a[k]*x^k, k = 1 .. 100): S := [seq(numtheory[mobius](i+1), i = 1 .. 100)]: Sabs := [seq(i, i = 1 .. 100)]: seq(assign(a[i] = solve(coeff(L, x^i) = `if`(is(i in Sabs), S[Search(i, Sabs)], 0), a[i])), i = 1 .. 100): U := [seq(a[i], i = 1 .. 100)]
# alternative:
N:= 100: # to get a(1) to a(N)
P[0]:= 1:
for n from 1 to N do
a[n]:= coeff(P[n-1], x, n) - numtheory:-mobius(n+1);
P[n]:= P[n-1]*(1-a[n]*x^n);
od:
seq(a[n], n=1..N); # Robert Israel, Jan 06 2016
MATHEMATICA
Module[{a, n = 15}, Array[a, n] /. Flatten@Solve[CoefficientList[Product[1 - a[i] x^i, {i, n}], x][[;; n + 1]] == Array[MoebiusMu, n + 1], Array[a, n]]] (* JungHwan Min, Jan 10 2016 *)
Module[{a, n = 25}, Array[a, n] /. Flatten@Solve[Table[Plus @@ Times @@@ Replace[Select[IntegerPartitions[m], DuplicateFreeQ], k_ :> -a[k], {2}] == MoebiusMu[m + 1], {m, n}], Array[a, n]]] (* JungHwan Min, Jan 10 2016 *)
P= 1; a[0] = 0; a[n_] := a[n] = Coefficient[P = Collect[P (1 - a[n - 1] x^(n - 1)), x], x, n] - MoebiusMu[n + 1]; Array[a, 40] (* JungHwan Min, Jan 17 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gevorg Hmayakyan, Jan 06 2016
STATUS
approved