A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

2, -3, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986

Offset: 1

Author

Christian G. Bower, Jan 04 1999

Keywords

• sign

Comments

Apart from initial terms, exponents in expansion of A065472 as a product zeta(n)^(-a(n)).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..3000
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• N. J. A. Sloane, Euler transform.

Crossrefs

Cf. A001037, A008683, A038064, A038065, A038066, A038067, A038068, A038069, A038070, A059966, A060477, A065472.

Programs

Mathematica

a[n_] := DivisorSum[n, (-1)^(#+1) * MoebiusMu[n/#]*2^# &] / n; Array[a, 33] (* Amiram Eldar, May 29 2025 *)

PARI

{a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

Formula

a(n) = (1/n) * Sum_{d divides n} (-1)^(d+1)*moebius(n/d)*2^d. - Vladeta Jovovic, Sep 06 2002

G.f.: Sum_{n>=1} moebius(n)*log(1 + 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010

For n == 0, 1, 3 (mod 4), a(n) = (-1)^(n+1)*A001037(n), which for n>1 also equals (-1)^(n+1)*A059966(n) = (-1)^(n+1)*A060477(n).

For n == 2 (mod 4), a(n) = -(A001037(n) + A001037(n/2)). - George Beck and Max Alekseyev, May 23 2016

a(n) ~ -(-1)^n * 2^n / n. - Vaclav Kotesovec, Jun 12 2018