A266949 a(n) is defined by Product_{i>=1} (1-a(i)*x^i) = Sum_{i>=0} möbius(i+1)*x^i.

1, 1, 1, 2, 2, 4, 6, 8, 13, 22, 35, 48, 89, 129, 228, 345, 609, 897, 1624, 2421, 4295, 6598, 11855, 18217, 32396, 49787, 88387, 139517, 246442, 380905, 684682, 1082651, 1895821, 3009692, 5346768, 8514026, 15024307, 23891567, 42093993, 68125683, 119570322

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

Cf. A008683.

Sequence in context: A274155 A145465 A291055 * A255710 A329137 A239851

Adjacent sequences: A266946 A266947 A266948 * A266950 A266951 A266952

Keyword

nonn

Author

Gevorg Hmayakyan, Jan 06 2016

Status

approved