A321875 a(n) = Sum_{d|n} d*d!.

1, 5, 19, 101, 601, 4343, 35281, 322661, 3265939, 36288605, 439084801, 5748023639, 80951270401, 1220496112085, 19615115520619, 334764638530661, 6046686277632001, 115242726706374263, 2311256907767808001, 48658040163569088701, 1072909785605898275299

Offset: 1

Author

Ilya Gutkovskiy, Nov 20 2018

Keywords

• nonn

Comments

Inverse Möbius transform of A001563.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..448
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000142, A001563, A062363, A107895.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k*Factorial(k)*x^k/(1 - x^k): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 20 2018

Mathematica

Table[Sum[d d!, {d, Divisors[n]}], {n, 21}]
nmax = 21; Rest[CoefficientList[Series[Sum[k k! x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 21; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

PARI

a(n) = sumdiv(n, d, d*d!); \\ Michel Marcus, Nov 20 2018

Sage

s = sum(k*factorial(k)*x^k/(1-x^k) for k in (1..24));
(s/x).series(x, 21).coefficients(x, sparse=false) # Peter Luschny, Nov 21 2018

Formula

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

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

a(n) = Sum_{d|n} A001563(d).