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%I #9 Sep 15 2020 21:28:29
%S 1,-1,7,-170,9664,-1080824,207876968,-63709383408,29068641741312,
%T -18924533538121728,16870738405288439808,-20048074289311310521344,
%U 30889296893650981899202560,-60580966918820974514054369280,148238116513927185591120536580096
%N a(0) = 1; a(n) = -(n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.
%F Sum_{n>=0} a(n) * x^n / (n!)^3 = 1 / (1 + polylog(3,x)).
%t a[0] = 1; a[n_] := a[n] = -(n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
%t nmax = 14; CoefficientList[Series[1/(1 + PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3
%o (PARI) a(n)={n!^3*polcoef(1/(1 + polylog(3,x + O(x*x^n))), n)} \\ _Andrew Howroyd_, Sep 15 2020
%Y Cf. A006252, A074706, A212856, A336259, A337677, A337678.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Sep 15 2020