A281904 Expansion of Sum_{i>=1} mu(i)^2*i*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j), where mu() is the Moebius function (A008683).

1, 4, 9, 16, 31, 58, 93, 144, 221, 343, 502, 733, 1048, 1495, 2089, 2881, 3947, 5357, 7205, 9618, 12758, 16812, 22001, 28623, 37037, 47720, 61121, 77973, 99029, 125322, 157874, 198205, 247954, 309203, 384260, 476116, 588149, 724613, 890175, 1090781, 1333193, 1625702, 1977505, 2400221, 2906800, 3513121

Offset: 1

Author

Ilya Gutkovskiy, Feb 01 2017

Keywords

• nonn

Comments

Total sum of squarefree parts in all partitions of n.

Convolution of the sequences A000041 and A048250.

Examples

			a(4) = 16 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1] and 3 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 16.
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A005117, A008683, A048250, A066186, A073118.

Programs

Mathematica

nmax = 46; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 i x^i/(1 - x^i), {i, 1, nmax}] / Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} mu(i)^2*i*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).