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%I #8 Sep 08 2018 06:04:47
%S 1,1,8,93,1544,32615,843264,25739539,906373376,36163950849,
%T 1612483625600,79458277381901,4288069172500992,251520785449249927,
%U 15932801526165085184,1084003570689331039875,78835487923639854792704,6103175938145968656408641,501114006272655771562911744
%N a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k)^(n/k).
%H Vaclav Kotesovec, <a href="/A299034/b299034.txt">Table of n, a(n) for n = 0..300</a>
%F 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).
%F a(n) ~ c * d^n * n^n, where d = 1.7257974131308983723949107467... and c = 0.693704376971941705824592525... - _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), 3, 11, 59, 339, 2629, ...
%e n = 2: 1, 2, (8), 40, 260, 1928, 17056, ...
%e n = 3: 1, 3, 15, (93), 711, 6237, 62901, ...
%e n = 4: 1, 4, 24, 176, (1544), 15456, 174784, ...
%e n = 5: 1, 5, 35, 295, 2915, (32615), 407725, ...
%e n = 6: 1, 6, 48, 456, 5004, 61704, (843264), ...
%t Table[n! SeriesCoefficient[Product[1/(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]
%Y Cf. A000005, A028342, A255672, A299033.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 01 2018