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%I M1672 N0658 #51 Sep 26 2022 20:33:04
%S 2,-6,24,-80,450,-2142,17696,-112464,1232370,-9761510,132951192,
%T -1258797696,20476388114,-225380451870,4261074439680,-53438049741152,
%U 1151146814425506,-16199301256675974,391615698778725080,-6109914386833902960
%N Logarithmic numbers.
%C From _Peter Bala_, Sep 06 2022: (Start)
%C Conjectures: Let k be a positive integer.
%C 1) for n >= 1, a(n+2*k) - a(n) is divisible by 2*k; if true, then the reduction of the sequence modulo 2*k gives a periodic sequence with period dividing 2*k.
%C 2) for n >= 1, a(n+2*k+1) + a(n) is divisible by 2*k+1; if true, then the reduction of the sequence modulo 2*k+1 gives a periodic sequence with period dividing 4*k + 2. (End)
%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="/A002742/b002742.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 <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>
%F E.g.f.: (2*x/(1-x^2)+log(1-x^2))*exp(-x). - _Sean A. Irvine_, Aug 11 2014
%F a(n) = 2*A002747(n) - a(n-1). - _R. J. Mathar_, Jul 24 2015
%F From _Emanuele Munarini_, Dec 16 2017: (Start)
%F a(n) = (-1)^(n-1)*Sum_{k=0..n} binomial(n+1,2*k+1)*((n-2*k)/(k+1))*(2*k+1)!.
%F a(n+3)+a(n+2)-(n+2)*(n+3)*a(n+1)-(n+2)*(n+3)*a(n) = 2*(-1)^n*(n+3).
%F (n+3)*a(n+4)+(2*n+7)*a(n+3)-(n+2)*(n+4)^2*a(n+2)-(n+3)*(n+4)*(2*n+5)*a(n+1)-(n+2)*(n+3)*(n+4)*a(n) = 0.
%F E.g.f.: A(x) = - D(exp(-x)*log(1-x^2)), where D is the derivative with respect to x. (End)
%F a(n) ~ n! * (exp(-1) - (-1)^n * exp(1)). - _Vaclav Kotesovec_, Dec 16 2017
%t Table[(-1)^(n-1)Sum[Binomial[n+1,2k+1](n-2k)/(k+1)(2k+1)!,{k,0,n}],{n,0,100}] (* _Emanuele Munarini_, Dec 16 2017 *)
%o (Maxima) makelist((-1)^(n-1)*sum(binomial(n+1,2*k+1)*(n-2*k)/(k+1)*(2*k+1)!,k,0,n),n,0,12); /* _Emanuele Munarini_, Dec 16 2017 */
%o (PARI) first(n) = x='x+O('x^(n+1)); Vec(serlaplace((2*x/(1-x^2)+log(1-x^2))*exp(-x))) \\ _Iain Fox_, Dec 16 2017
%Y Cf. A002741.
%K sign
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Jeffrey Shallit_
%E More terms from _Sean A. Irvine_, Aug 11 2014
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