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%I A299033 #5 Feb 01 2018 20:57:16
%S A299033 1,-1,0,15,-136,885,-4896,43085,-787200,7775271,326355200,
%T A299033 -22138191801,781498160640,-18924340012435,239123351330304,
%U A299033 5915023788331125,-568462201562300416,25327272129182225295,-795994018378027868160,15538852668590468027711
%N A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).
%F A299033 a(n) = n! * [x^n] exp(-n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005).
%e A299033 The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 - x^k)^(n/k) begins:
%e A299033 n = 0: (1),  0,    0,    0,     0,      0,      0,  ...
%e A299033 n = 1:  1, (-1),  -1,    1,    -1,     41,   -131,  ...
%e A299033 n = 2:  1,  -2,   (0),   8,    -4,     72,   -704,  ...
%e A299033 n = 3:  1,  -3,    3,  (15),  -45,     63,  -1539,  ...
%e A299033 n = 4:  1,  -4,    8,   16, (-136),   224,  -1856,  ...
%e A299033 n = 5:  1,  -5,   15,    5,  -265,   (885), -2075,  ...
%e A299033 n = 6:  1,  -6,   24,  -24,  -396,   2376, (-4896), ...
%t A299033 Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
%Y A299033 Cf. A000005, A028343, A281267, A299034.
%K A299033 sign
%O A299033 0,4
%A A299033 _Ilya Gutkovskiy_, Feb 01 2018

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