A321438 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^n.

1, 3, 28, 239, 3126, 45990, 823544, 16711423, 387440173, 9990235398, 285311670612, 8913939907598, 302875106592254, 11111328602501550, 437893920912786408, 18446462594437808127, 827240261886336764178, 39346258082220810086373, 1978419655660313589123980, 104857499999905732078938574

Offset: 1

Author

Ilya Gutkovskiy, Nov 09 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A000593, A023887, A078306, A078307, A186633, A284900, A284926, A284927, A321385.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(k*x)^k/(1+(k*x)^k): k in [1..m]]) ));  // G. C. Greubel, Nov 11 2018

Mathematica

Table[Sum[(-1)^(n/d + 1) d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[(k x)^k/(1 + (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 20; Rest[CoefficientList[Series[Log[Product[(1 + k^k x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

PARI

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^n); \\ Michel Marcus, Nov 09 2018

Formula

G.f.: Sum_{k>=1} (k*x)^k/(1 + (k*x)^k).

L.g.f.: log(Product_{k>=1} (1 + k^k*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.

a(n) ~ n^n. - Vaclav Kotesovec, Nov 10 2018