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%I #9 Sep 08 2018 06:05:27
%S 1,-1,4,-39,536,-9115,185904,-4461877,123647488,-3886461081,
%T 136538590400,-5300491027711,225313697972736,-10409021924850211,
%U 519298241645107456,-27824560148201248125,1593597443825288904704,-97153909607626767338353,6281720886474120790582272
%N a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).
%H Vaclav Kotesovec, <a href="/A300188/b300188.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = n! * [x^n] exp(-n*Sum_{k>=1} A048272(k)*x^k/k).
%F a(n) ~ (-1)^n * c * d^n * n^n, where d = 1.3587950730244927060955... and c = 0.6449711831436950784... - _Vaclav Kotesovec_, Sep 08 2018
%e The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
%e n = 0: (1), 0, 0, 0, 0, 0, 0, ...
%e n = 1: 1, (-1), 1, -5, 23, -119, 619, ...
%e n = 2: 1, -2, (4), -16, 92, -568, 3856, ...
%e n = 3: 1, -3, 9, (-39), 243, -1737, 13671, ...
%e n = 4: 1, -4, 16, -80, (536), -4256, 37504, ...
%e n = 5: 1, -5, 25, -145, 1055, (-9115), 88075, ...
%e n = 6: 1, -6, 36, -240, 1908, -17784, (185904), ...
%t Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]
%Y Cf. A048272, A281266, A294356, A299033, A299034, A300187.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Feb 28 2018