A002747 Logarithmic numbers. (Formerly M1924 N0759)

1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302

Offset

1,2

Comments

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

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

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

Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

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

Simon Plouffe, Simple inverter lookup on 1.8134302039235

Simon Plouffe, Smart inverter lookup on 1.8134302039235

Index entries for sequences related to logarithmic numbers

Formula

E.g.f.: x/exp(x)/(1-x^2). - Vladeta Jovovic, Feb 09 2003

a(n) = n*((n-1)*a(n-2)-(-1)^n). - Matthew Vandermast, Jun 30 2003

From Gerald McGarvey, Jun 06 2004: (Start)

For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.

For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.

For n odd, lim_{n->infinity} a(n)/n! = cosh(1).

For n even, lim_{n->infinity} a(n)/n! = sinh(1).

For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).

For signed values, Sum_{n>=1} a(n)/n!^2 = 0.

For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)

a(n) = (-1)^(n-1)*Sum_{k=0..n} C(n, k)*k!*(1-(-1)^k)/2. - Paul Barry, Sep 14 2004

a(n) = (-1)^(n+1)*n*A087208(n-1). - R. J. Mathar, Jul 24 2015

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

Maple

a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2013

Mathematica

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 *)
a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 1, 22}] (* Peter Luschny, Dec 18 2017 *)

Prog

(PARI) a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022

Crossrefs

Cf. A000166, A000522, A087208.

Sequence in context: A324372 A138912 A374827 * A110377 A338436 A041877

Adjacent sequences: A002744 A002745 A002746 * A002748 A002749 A002750

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

Status

approved