A266949 - OEIS

%I #42 Nov 13 2016 15:59:45

%S 1,1,1,2,2,4,6,8,13,22,35,48,89,129,228,345,609,897,1624,2421,4295,

%T 6598,11855,18217,32396,49787,88387,139517,246442,380905,684682,

%U 1082651,1895821,3009692,5346768,8514026,15024307,23891567,42093993,68125683,119570322

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

%C Conjecture: All coefficients are positive and strictly increasing starting from 5th term.

%C 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, ...].

%H Robert Israel, <a href="/A266949/b266949.txt">Table of n, a(n) for n = 1..1000</a>

%p 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)]

%p # alternative:

%p N:= 100: # to get a(1) to a(N)

%p P[0]:= 1:

%p for n from 1 to N do

%p a[n]:= coeff(P[n-1],x,n) - numtheory:-mobius(n+1);

%p P[n]:= P[n-1]*(1-a[n]*x^n);

%p od:

%p seq(a[n],n=1..N); # _Robert Israel_, Jan 06 2016

%t 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 *)

%t 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 *)

%t 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 *)

%Y Cf. A008683.

%K nonn

%O 1,4

%A _Gevorg Hmayakyan_, Jan 06 2016