|
%I M1924 N0759 #94 Sep 26 2022 20:33:22
%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 abs(a(n)) is also the number of distinct routes starting from a point A and ending at a point B, without traversing any edge more than once, when there are n bi-directional edges connecting A and B. E.g., if there are 3 edges p, q and r from A to B, then the 9 routes starting from A and ending at B are p, q, r, pqr, prq, rpq, rqp, qpr and qrp. - _Nikita Kiran_, Sep 02 2022
%C Reducing the sequence modulo the odd integer 2*k + 1 results in a purely periodic sequence with period dividing 4*k + 2, For example, reduced modulo 5 the sequence becomes the purely periodic sequence [1, 3, 4, 2, 0, 4, 2, 1, 3, 0, 1, 3, 4, 2, 0, 4, 2, 1, 3, 0, ...] with period 10. - _Peter Bala_, Sep 12 2022
%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 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</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&lookup_type=simple">Inverter lookup on 1.8134302039235</a> [Broken link]
%H Simon Plouffe, <a href="https://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=1.8134302039235&lookup_type=smart">Smart 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 From _Gerald McGarvey_, Jun 06 2004: (Start)
%F For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.
%F For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.
%F For n odd, lim_{n->infinity} a(n)/n! = cosh(1).
%F For n even, lim_{n->infinity} a(n)/n! = sinh(1).
%F For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).
%F For signed values, Sum_{n>=1} a(n)/n!^2 = 0.
%F For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)
%F a(n) = (-1)^(n-1)*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-François 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 *)
%o (PARI) a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ _Michel Marcus_, Jan 13 2022
%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
|
