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A002747 Logarithmic numbers.
(Formerly M1924 N0759)
5

%I M1924 N0759

%S 1,-2,9,-28,185,-846,7777,-47384,559953,-4264570,61594841,-562923252,

%T 9608795209,-102452031878,2017846993905,-24588487650736,

%U 548854382342177,-7524077221125234,187708198761024553,-2859149344027588940,78837443479630312281,-1320926996940746090302

%N Logarithmic numbers.

%C For n odd, lim n->inf a(n)/n! = cosh(1). For n even, lim n->inf a(n)/n! = sinh(1) lim n->inf n*a(n)*a(n-1)/n!^2 = cosh(1)sinh(1). For signed values, Sum n=1..inf a(n)/n!^2 = 0. For unsigned values, Sum n=1..inf a(n)/n!^2 = cosh(1)sinh(1). - _Gerald McGarvey_, Jun 06 2004

%D J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002747/b002747.txt">Table of n, a(n) for n = 1..200</a>

%H J. M. Gandhi, <a href="/A002741/a002741.pdf">On logarithmic numbers</a>, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]

%H Simon Plouffe, <a href="http://oldweb.cecm.sfu.ca/cgi-bin/isc/lookup?number=1.8134302039235&amp;lookup_type=simple">Inverter lookup on 1.8134302039235</a>

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F E.g.f.: x/exp(x)/(1-x^2). - _Vladeta Jovovic_, Feb 09 2003

%F a(n) = n*((n-1)*a(n-2)-(-1)^n). - _Matthew Vandermast_, Jun 30 2003

%F For n odd, n! * Sum_{i=0..n-1 i even} 1/i!, for n even, n! * Sum_{i=1..n-1, i odd} 1/i!. - _Gerald McGarvey_, Jun 06 2004

%F a(n) = (-1)^n*sum{k=0..n, C(n, k)k!(1-(-1)^k)/2}. - _Paul Barry_, Sep 14 2004

%F a(n) = (-1)^(n+1)*n*A087208(n-1). - _R. J. Mathar_, Jul 24 2015

%F a(n) = (exp(-1)*Gamma(1+n,-1) - (-1)^n*exp(1)*Gamma(1+n,1))/2 = (A000166(n) - (-1)^n*A000522(n))/2. - _Peter Luschny_, Dec 18 2017

%p a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:

%p seq(a(n), n=1..25); # _Alois P. Heinz_, Jul 10 2013

%t egf = x/Exp[x]/(1-x^2); a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!; Table[a[n], {n, 1, 22}] (* _Jean-Fran├žois Alcover_, Jan 17 2014, after _Vladeta Jovovic_ *)

%t a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;

%t Table[a[n], {n, 1, 22}] (* _Peter Luschny_, Dec 18 2017 *)

%Y Cf. A000166, A000522, A087208.

%K sign

%O 1,2

%A _N. J. A. Sloane_.

%E More terms from _Jeffrey Shallit_.

%E More terms from _Vladeta Jovovic_, Feb 09 2003

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)