A308290 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k)^3)/k.

1, 2, 3, 1, -1, -6, -3, 2, 9, 9, -6, -24, -25, 16, 72, 75, -35, -213, -239, 78, 627, 767, -182, -1890, -2477, 355, 5847, 8109, -360, -18195, -26801, -1225, 56724, 89040, 11431, -177897, -297030, -61857, 560310, 994427, 284075, -1766754, -3338212, -1201932

Offset

1,2

Comments

Inverse Euler transform of triangular numbers (A000217).

Links

Table of n, a(n) for n=1..44.

Formula

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000217.

Mathematica

nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - x^k)^3]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j^2*(j + 1)/2 - Sum[s[[d]]*(j - d)*(j - d + 1)/2, {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 *)

Crossrefs

Cf. A000217, A000294, A008683, A308291, A316152.

Sequence in context: A342673 A365463 A068348 * A204167 A217897 A135900

Adjacent sequences: A308287 A308288 A308289 * A308291 A308292 A308293

Keyword

sign

Author

Ilya Gutkovskiy, May 18 2019

Status

approved