A308447 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/((1 - x^k)*(1 - 2*x^k)))/k.

1, 2, 4, 5, 8, 8, 16, 25, 52, 98, 192, 345, 640, 1162, 2164, 4050, 7680, 14534, 27648, 52479, 99956, 190554, 364544, 698525, 1341848, 2580790, 4971616, 9587565, 18513920, 35790276, 69271552, 134211600, 260297012, 505286430, 981714296, 1908881520, 3714580480, 7233615306

Offset: 1

Author

Ilya Gutkovskiy, May 27 2019

Keywords

• nonn

Comments

Inverse Euler transform of A000225.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000225, A001037, A008683, A059966, A319918.

Programs

Mathematica

nmax = 38; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/((1 - x^k) (1 - 2 x^k))]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2^ j - 1) - Sum[s[[d]]*(2^(j - d) - 1), {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 10 2019 *)

Formula

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000225.

a(n) ~ 2^n/n. - Vaclav Kotesovec, May 28 2019