A308536 - OEIS

%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