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A383897 Expansion of e.g.f. log(1 + x)/(1 - 2*x).

%I A383897 #49 May 29 2025 00:52:08
%S A383897 0,1,3,20,154,1564,18648,261792,4183632,75345696,1506551040,
%T A383897 33147751680,795506123520,20683638213120,579135642946560,
%U A383897 17374156466688000,555971699259648000,18903058697617920000,680509757426817024000,25859377184592752640000,1034374965738609696768000
%N A383897 Expansion of e.g.f. log(1 + x)/(1 - 2*x).
%H A383897 Vincenzo Librandi, <a href="/A383897/b383897.txt">Table of n, a(n) for n = 0..200</a>
%F A383897 a(n) = n! * Sum_{k=1..n} (-1)^(k-1) * 2^(n-k)/k.
%F A383897 a(n) = 2 * n * a(n-1) - (-1)^n * (n-1)!.
%F A383897 a(n) = (n+1) * a(n-1) + 2 * (n-1)^2 * a(n-2).
%F A383897 a(n) ~ log(3/2) * 2^n * n!. - _Vaclav Kotesovec_, May 23 2025
%F A383897 a(n) = n!*((-1)^(n+1)/n + (-1)^n*LerchPhi(-1/2,1,n) + 2^n*log(3/2)) for n > 0. - _Stefano Spezia_, May 23 2025
%t A383897 a[n_]:= n! * Sum[(-1)^(k-1)*2^(n-k)/k,{k,1,n}];Table[a[n],{n,0,19}] (* _Vincenzo Librandi_, May 23 2025 *)
%o A383897 (PARI) a(n) = n!*sum(k=1, n, (-1)^(k-1)*2^(n-k)/k);
%o A383897 (Magma) [0] cat [n le 1 select 1  else  2*n * Self(n-1) - (-1)^n * Factorial(n-1): n in [1..20]]; // _Vincenzo Librandi_, May 23 2025
%Y A383897 Cf. A024167, A384199.
%Y A383897 Cf. A068102.
%K A383897 nonn,easy
%O A383897 0,3
%A A383897 _Seiichi Manyama_, May 22 2025