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%I #45 Sep 08 2022 08:46:08
%S 0,2,15,128,1310,15864,222936,3572736,64354608,1287495360,28328889600,
%T 679936896000,17678878214400,495015296025600,14850552286080000,
%U 475219068007219200,16157470542709708800,581669316147767500800,22103440771676298854400
%N a(n) = r * (n-1)! where r is the rational number that satisfies the equation Sum_{k>=n} (-1)^(k + n)/C(k,n) = n*2^(n-1)*log(2) - r.
%C The sum of the terms of the inverse of the binomial coefficients, 1/C(k,n), with alternating signs, equals an irrational number which is expressed as m * log(2) - r, where m is the integer n*2^(n-1) = A001787(n), n>=1, and r is rational. a(n) = r * (n-1)!.
%H G. C. Greubel, <a href="/A242091/b242091.txt">Table of n, a(n) for n = 1..400</a>
%F From _Robert Israel_, Aug 14 2014: (Start)
%F a(n) = n * A068102(n-1).
%F a(n) = n! * Sum_{j=1..(n-1)} 2^(n-j-1)/j).
%F a(n) = n! * (2^(n-1)*log(2)-(1/2)*LerchPhi(1/2, 1, n)).
%F a(n+1) = 2*(n+1)*a(n) + (n+1)!/n.
%F E.g.f.: x*log(1-x)/(2*x-1).
%F (End)
%F Recurrence: (n-1)*a(n) = n*(3*n-4)*a(n-1) - 2*(n-2)*(n-1)*n*a(n-2). - _Vaclav Kotesovec_, Aug 15 2014
%e Sum_{k>=1} (-1)^(k + 1)/C(k,1) = Sum_{k>=1} (-1)^(k + 1)/k = log(2) where m = 1 and r = 0. (See A002162.)
%e Sum_{k>=2} (-1)^(k + 2)/C(k,2) = 4*log(2) - 2. (See A000217.)
%e Sum_{k>=3} (-1)^(k + 3)/C(k,3) = 12*log(2) - 15/2. (See A000292.)
%e Sum_{k>=4} (-1)^(k + 4)/C(k,4) = 30*log(2) - 64/3. (See A000332.)
%e Sum_{k>=5} (-1)^(k + 5)/C(k,5) = 80*log(2) - 655/12. (See A000389.)
%p seq(add(2^(n-j-1)*n!/j, j=1..n-1), n=1..100); # _Robert Israel_, Aug 14 2014
%t Table[Sum[2^(n - j - 1)*n!/j, {j, n - 1}], {n, 20}] (* _Wesley Ivan Hurt_, Aug 14 2014 *)
%t FullSimplify[Table[-1/2*n!*(LerchPhi[1/2, 1, n] - 2^n*Log[2]),{n, 1, 20}]] (* _Vaclav Kotesovec_, Aug 15 2014 *)
%o (Magma) [n le 1 select 0 else 2*(n)*Self(n-1)+(Factorial(n) div (n-1)): n in [1..20]]; // _Vincenzo Librandi_, Sep 22 2015
%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(x*log(1-x)/(2*x-1)))) \\ _G. C. Greubel_, Nov 25 2017
%Y Cf. A007318, A242023, A242024, A000217, A000292, A000332, A000389, A000579, A000580, etc. and A002162, A001787, A068102.
%K nonn
%O 1,2
%A _Richard R. Forberg_, Aug 14 2014
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