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a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k).
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%I #4 Jul 12 2020 12:18:05

%S 1,-1,3,-10,-117,5224,-23010,-10319891,463834315,69461529092,

%T -10005601418172,-1323175060249241,468450359815048182,

%U 63281374513705043227,-46495538420749056681263,-7147072328212024308730535,9119277358213513566069911755,1827085356172328516064256064092

%N a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k).

%F a(n) = (n!)^3 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^3).

%t a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^3 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]

%t nmax = 17; CoefficientList[Series[Exp[-Sum[x^k/(k!)^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3

%Y Cf. A000587, A061684, A336209.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jul 12 2020