%I #10 Sep 05 2023 05:35:32

%S 1,4,6,4,10,24,14,0,27,40,22,24,26,56,60,-16,34,108,38,40,84,88,46,0,

%T 75,104,108,56,58,240,62,-64,132,136,140,108,74,152,156,0,82,336,86,

%U 88,270,184,94,-96,147,300,204,104,106,432,220,0,228,232,118,240,122,248,378,-192

%N Expansion of g.f. Sum_{k>0} k * x^k / (1 + (-x)^k)^2.

%F a(n) is multiplicative with a(2^e) = (3-e) * 2^e if e>0, a(p^e) = (e+1) * p^e if p>2.

%F a(16*n + 8) = 0.

%e x + 4*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 24*x^6 + 14*x^7 + 27*x^9 + 40*x^10 + ...

%t f[p_, e_] := (e+1) * p^e; f[2, e_] := (3-e) * 2^e; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Sep 05 2023 *)

%o (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, (3-e), e+1) * p^e)))}

%o (PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k * x^k / (1 + (-x)^k)^2, x*O(x^n)), n))}

%Y A038040(2*n + 1) = a(2*n + 1). -16 * A038040(n) = a(16*n).

%K sign,easy,mult

%O 1,2

%A _Michael Somos_, Aug 22 2008