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A002747 Logarithmic numbers.
(Formerly M1924 N0759)
5
1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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]

Simon Plouffe, 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

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

a(n) = (-1)^n*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-Fran├žois 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 *)

CROSSREFS

Cf. A000166, A000522, A087208.

Sequence in context: A291632 A324372 A138912 * A110377 A041877 A090208

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

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Last modified August 25 05:19 EDT 2019. Contains 326318 sequences. (Running on oeis4.)