A143521 Expansion of g.f. Sum_{k>0} k * x^k / (1 + (-x)^k)^2.

1, 4, 6, 4, 10, 24, 14, 0, 27, 40, 22, 24, 26, 56, 60, -16, 34, 108, 38, 40, 84, 88, 46, 0, 75, 104, 108, 56, 58, 240, 62, -64, 132, 136, 140, 108, 74, 152, 156, 0, 82, 336, 86, 88, 270, 184, 94, -96, 147, 300, 204, 104, 106, 432, 220, 0, 228, 232, 118, 240, 122, 248, 378, -192

Offset

1,2

Links

Table of n, a(n) for n=1..64.

Formula

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.

a(16*n + 8) = 0.

Example

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 + ...

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A328045 A277278 A328722 * A278363 A123969 A255679

Adjacent sequences: A143518 A143519 A143520 * A143522 A143523 A143524

Keyword

sign,easy,mult

Author

Michael Somos, Aug 22 2008

Status

approved