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A002741 Logarithmic numbers: expansion of the e.g.f. -log(1-x) * e^(-x).

%I A002741 M0037 N0010 #60 Nov 28 2022 18:51:04
%S A002741 0,1,-1,2,0,9,35,230,1624,13209,120287,1214674,13469896,162744945,
%T A002741 2128047987,29943053062,451123462672,7245940789073,123604151490591,
%U A002741 2231697509543362,42519034050101744,852495597142800377,17942811657908144163,395553947953212635718,9114871523102565301544,219135339782236105192745
%N A002741 Logarithmic numbers: expansion of the e.g.f. -log(1-x) * e^(-x).
%D A002741 J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
%D A002741 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002741 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002741 T. D. Noe, <a href="/A002741/b002741.txt">Table of n, a(n) for n = 0..100</a>
%H A002741 J. M. Gandhi, <a href="/A002741/a002741.pdf">On logarithmic numbers</a>, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
%H A002741 <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>
%F A002741 From _Michael Somos_, Jun 21 2002: (Start)
%F A002741 E.g.f.: -log(1-x) / e^x.
%F A002741 a(n) = (n-2) * a(n-1) + (n-1) * a(n-2) - (-1)^n, n > 0.
%F A002741 A000757(n) = (-1)^n + a(n). (End)
%F A002741 a(n) = n-th forward difference of [0, 1, 1, 2, 6, 24, ...] (factorials A000142 with 0 prepended). - _Michael Somos_, Mar 28 2011
%F A002741 a(n) ~ exp(-1)*(n-1)!. - _Vaclav Kotesovec_, Mar 10 2014
%F A002741 From _Vladimir Reshetnikov_, Oct 29 2015: (Start)
%F A002741 Recurrence: a(0) = 0, a(1) = 1, a(2) = -1, a(n) = (n-3)*a(n-1) + 2*(n-2)*a(n-2) + (n-2)*a(n-3).
%F A002741 a(n) + a(n+1) = A000166(n). (End)
%F A002741 a(n) = (-1)^(n-1)*n*hypergeom([1,1,1-n], [2], 1). - _Peter Luschny_, May 09 2017
%e A002741 a(3) = 2 = 2! - 3*1! + 3*0! - 0. a(4) = 0 = 3! - 4*2! + 6*1! - 4*0! + 0. - _Michael Somos_, Mar 28 2011
%p A002741 a := n -> (-1)^(n-1)*n*hypergeom([1,1,1-n], [2], 1):
%p A002741 seq(simplify(a(n)), n = 0..25); # _Peter Luschny_, May 09 2017
%t A002741 a[n_] := Sum[(-1)^k*n!/((n-k)*k!), {k, 0, n-1}]; Table[a[n], {n, 0, 19}](* _Jean-François Alcover_, Nov 21 2011 *)
%t A002741 With[{nn=30},CoefficientList[Series[-Log[1-x] E^-x,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 28 2022 *)
%o A002741 (PARI) {a(n) = if( n<0, 0, sum( k=0, n-1, (-1)^k * binomial(n, k) * (n - k - 1)!))} /* _Michael Somos_, Jun 21 2002 */
%Y A002741 Cf. A000142, A000166, A000757.
%K A002741 sign,easy,nice
%O A002741 0,4
%A A002741 _N. J. A. Sloane_
%E A002741 More terms from _Jeffrey Shallit_
%E A002741 More terms from _Joerg Arndt_, Sep 02 2013