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

%I A321875 #16 Sep 08 2022 08:46:23
%S A321875 1,5,19,101,601,4343,35281,322661,3265939,36288605,439084801,
%T A321875 5748023639,80951270401,1220496112085,19615115520619,334764638530661,
%U A321875 6046686277632001,115242726706374263,2311256907767808001,48658040163569088701,1072909785605898275299
%N A321875 a(n) = Sum_{d|n} d*d!.
%C A321875 Inverse Möbius transform of A001563.
%H A321875 Seiichi Manyama, <a href="/A321875/b321875.txt">Table of n, a(n) for n = 1..448</a>
%H A321875 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A321875 G.f.: Sum_{k>=1} k*k!*x^k/(1 - x^k).
%F A321875 L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k!)) = Sum_{n>=1} a(n)*x^n/n.
%F A321875 a(n) = Sum_{d|n} A001563(d).
%t A321875 Table[Sum[d d!, {d, Divisors[n]}], {n, 21}]
%t A321875 nmax = 21; Rest[CoefficientList[Series[Sum[k k! x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A321875 nmax = 21; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%o A321875 (PARI) a(n) = sumdiv(n, d, d*d!); \\ _Michel Marcus_, Nov 20 2018
%o A321875 (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 A321875 (Sage)
%o A321875 s = sum(k*factorial(k)*x^k/(1-x^k) for k in (1..24));
%o A321875 (s/x).series(x, 21).coefficients(x, sparse=false) # _Peter Luschny_, Nov 21 2018
%Y A321875 Cf. A000142, A001563, A062363, A107895.
%K A321875 nonn
%O A321875 1,2
%A A321875 _Ilya Gutkovskiy_, Nov 20 2018