A002742 Logarithmic numbers. (Formerly M1672 N0658)

2, -6, 24, -80, 450, -2142, 17696, -112464, 1232370, -9761510, 132951192, -1258797696, 20476388114, -225380451870, 4261074439680, -53438049741152, 1151146814425506, -16199301256675974, 391615698778725080, -6109914386833902960

Offset

1,1

Comments

From Peter Bala, Sep 06 2022: (Start)

Conjectures: Let k be a positive integer.

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.

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)

References

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

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..200

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]

Index entries for sequences related to logarithmic numbers

Formula

E.g.f.: (2*x/(1-x^2)+log(1-x^2))*exp(-x). - Sean A. Irvine, Aug 11 2014

a(n) = 2*A002747(n) - a(n-1). - R. J. Mathar, Jul 24 2015

From Emanuele Munarini, Dec 16 2017: (Start)

a(n) = (-1)^(n-1)*Sum_{k=0..n} binomial(n+1,2*k+1)*((n-2*k)/(k+1))*(2*k+1)!.

a(n+3)+a(n+2)-(n+2)*(n+3)*a(n+1)-(n+2)*(n+3)*a(n) = 2*(-1)^n*(n+3).

(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.

E.g.f.: A(x) = - D(exp(-x)*log(1-x^2)), where D is the derivative with respect to x. (End)

a(n) ~ n! * (exp(-1) - (-1)^n * exp(1)). - Vaclav Kotesovec, Dec 16 2017

Mathematica

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 *)

Prog

(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 */
(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

Crossrefs

Cf. A002741.

Sequence in context: A364819 A189858 A189568 * A048120 A322198 A003759

Adjacent sequences: A002739 A002740 A002741 * A002743 A002744 A002745

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Jeffrey Shallit

More terms from Sean A. Irvine, Aug 11 2014

Status

approved