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%I #7 Jun 29 2019 13:36:30
%S 1,1,3,14,86,664,6136,66240,816672,11331552,174662304,2961774144,
%T 54785368128,1097882522112,23693117756928,547844658441216,
%U 13511950038494208,354086653712228352,9824794572366544896,287752569360558907392,8871374335098501292032
%N Expansion of e.g.f. (1 - log(1 + x))/(1 - 2*log(1 + x)).
%C Inverse Stirling transform of A002866.
%F a(0) = 1; a(n) = Sum_{k=1..n} Stirling1(n,k) * 2^(k-1) * k!.
%F a(n) ~ n! * exp(1/2) / (4 * (exp(1/2) - 1)^(n+1)). - _Vaclav Kotesovec_, Jun 29 2019
%t nmax = 20; CoefficientList[Series[(1 - Log[1 + x])/(1 - 2 Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{1}, Table[Sum[StirlingS1[n, k] 2^(k - 1) k!, {k, 1, n}], {n, 1, 20}]]
%Y Cf. A002866, A008275, A011782, A050351, A088501, A306042, A308877.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 29 2019
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