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%I #7 Jun 06 2019 21:59:43
%S 1,-2,0,8,16,-64,-576,-1152,12800,136704,422912,-4464640,-72626176,
%T -413966336,1805123584,64448004096,651340611584,1132294045696,
%U -73000566390784,-1332193006190592,-10293724166750208,56984418960539648,3042980275005947904,46913652420264329216
%N Expansion of e.g.f. exp(1 - exp(2*x)).
%F O.g.f.: 1/(1 + 2*x/(1 - 2*x/(1 + 2*x/(1 - 4*x/(1 + 2*x/(1 - 6*x/(1 + 2*x/(1 - 8*x/(1 + ...))))))))), a continued fraction.
%F a(0) = 1; a(n) = -Sum_{k=1..n} 2^k*binomial(n-1,k-1)*a(n-k).
%F a(n) = exp(1) * 2^n * Sum_{k>=0} (-1)^k*k^n/k!.
%F a(n) = 2^n * A000587(n).
%t nmax = 23; CoefficientList[Series[Exp[1 - Exp[2x]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = -Sum[2^k Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%t Table[2^n BellB[n, -1], {n, 0, 23}]
%Y Cf. A000079, A000587, A009235, A055882, A213170.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Jun 06 2019