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A308536
Expansion of e.g.f. exp(1 - exp(2*x)).
3
1, -2, 0, 8, 16, -64, -576, -1152, 12800, 136704, 422912, -4464640, -72626176, -413966336, 1805123584, 64448004096, 651340611584, 1132294045696, -73000566390784, -1332193006190592, -10293724166750208, 56984418960539648, 3042980275005947904, 46913652420264329216
OFFSET
0,2
FORMULA
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.
a(0) = 1; a(n) = -Sum_{k=1..n} 2^k*binomial(n-1,k-1)*a(n-k).
a(n) = exp(1) * 2^n * Sum_{k>=0} (-1)^k*k^n/k!.
a(n) = 2^n * A000587(n).
MATHEMATICA
nmax = 23; CoefficientList[Series[Exp[1 - Exp[2x]], {x, 0, nmax}], x] Range[0, nmax]!
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}]
Table[2^n BellB[n, -1], {n, 0, 23}]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jun 06 2019
STATUS
approved