A348956 - OEIS

%I #11 Dec 14 2021 12:10:11

%S 1,0,-1,1,-1,1,0,1,-2,0,0,1,0,1,-1,-1,-3,1,3,1,1,0,-1,1,-2,0,-1,-2,-1,

%T 1,3,1,-2,0,2,0,2,1,-4,0,-1,1,1,1,0,-4,0,1,-6,1,2,-3,0,1,5,0,0,3,0,1,

%U 3,1,-4,-1,-3,0,3,1,3,-1,-1,1,0,1,-3,-4,-4,1,5,1,-4

%N a(0) = 1; a(n) = Sum_{d|n, d < n} (-1)^(n/d + 1) * a(d - 1).

%H Antti Karttunen, <a href="/A348956/b348956.txt">Table of n, a(n) for n = 0..20000</a>

%H Ilya Gutkovskiy, <a href="/A348956/a348956.jpg">Scatterplot of partial sums of A348956</a>

%F G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x^2) + x^3 * A(x^3) - x^4 * A(x^4) + ...

%t a[0] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^(n/d + 1) a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 80}]

%t nmax = 80; A[_] = 0; Do[A[x_] = 1 - Sum[(-x)^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]

%o (PARI) A348956(n) = if(!n,1,sumdiv(n,d,if(d<n,A348956(d-1)*(-1)^(1 + (n/d)),0))); \\ _Antti Karttunen_, Nov 05 2021

%Y Cf. A067856, A167865, A307776, A308077, A335062, A338639, A343371.

%K sign,look

%O 0,9

%A _Ilya Gutkovskiy_, Nov 04 2021