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Expansion of e.g.f. 1/(1 + x * log(1-2*x)).
2

%I #11 Jul 11 2024 13:40:50

%S 1,0,4,12,160,1440,20928,309120,5604352,110896128,2480762880,

%T 60669480960,1625897189376,47158671605760,1475038408015872,

%U 49434699759943680,1768135111433256960,67206058595340779520,2705431746964327759872

%N Expansion of e.g.f. 1/(1 + x * log(1-2*x)).

%F a(0) = 1; a(n) = n! * Sum_{k=2..n} 2^(k-1)/(k-1) * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * k! * |Stirling1(n-k,k)|/(n-k)!.

%t With[{nn=20},CoefficientList[Series[1/(1+x Log[1-2x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 11 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

%Y Cf. A367884.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 04 2023