A321875 - OEIS

%I #16 Sep 08 2022 08:46:23

%S 1,5,19,101,601,4343,35281,322661,3265939,36288605,439084801,

%T 5748023639,80951270401,1220496112085,19615115520619,334764638530661,

%U 6046686277632001,115242726706374263,2311256907767808001,48658040163569088701,1072909785605898275299

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

%C Inverse Möbius transform of A001563.

%H Seiichi Manyama, <a href="/A321875/b321875.txt">Table of n, a(n) for n = 1..448</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

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

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

%t Table[Sum[d d!, {d, Divisors[n]}], {n, 21}]

%t nmax = 21; Rest[CoefficientList[Series[Sum[k k! x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

%t nmax = 21; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

%o (PARI) a(n) = sumdiv(n, d, d*d!); \\ _Michel Marcus_, Nov 20 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k*Factorial(k)*x^k/(1 - x^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Nov 20 2018

%o (Sage)

%o s = sum(k*factorial(k)*x^k/(1-x^k) for k in (1..24));

%o (s/x).series(x, 21).coefficients(x, sparse=false) # _Peter Luschny_, Nov 21 2018

%Y Cf. A000142, A001563, A062363, A107895.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Nov 20 2018