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%I #34 Sep 19 2021 08:03:14
%S 1,1,8,6,144,480,5760,5040,524160,2177280,43545600,159667200,
%T 6706022400,49816166400,2092278988800,1307674368000,376610217984000,
%U 4623936565248000,128047474114560000,729870602452992000,77852864261652480000,613091306060513280000
%N a(n) = n! * Sum_{d in D(n+1)} (-1)^(d+1)*(n+1)/d, D(n) the divisors of n.
%F E.g.f.: d/dx log(Product_{k>=1} (1 + x^k)). - _Ilya Gutkovskiy_, Oct 15 2017
%F a(n) = n! * A000593(n+1). - _Seiichi Manyama_, Nov 08 2020.
%F E.g.f.: d/dx ( Sum_{k>=1} x^k / (k * (1 - x^(2*k))) ). - _Seiichi Manyama_, Sep 18 2021
%t Rest[CoefficientList[Series[Log[QPochhammer[-1, x]/2], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 15 2017 *)
%o (Sage)
%o A265024 = lambda n: factorial(n)*sum((-1)^(d+1)*(n+1)/d for d in divisors(n+1))
%o [A265024(n) for n in (0..21)]
%o (PARI) a(n) = n!*sumdiv(n+1, d, (-1)^(d+1)*(n+1)/d); \\ _Michel Marcus_, Jan 26 2016
%Y Cf. A000593, A027750, A038048, A075525 (Bell transform).
%K nonn
%O 0,3
%A _Peter Luschny_, Jan 26 2016
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