OFFSET
0,4
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
T. D. Noe, Table of n, a(n) for n = 0..100
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
FORMULA
From Michael Somos, Jun 21 2002: (Start)
E.g.f.: -log(1-x) / e^x.
a(n) = (n-2) * a(n-1) + (n-1) * a(n-2) - (-1)^n, n > 0.
A000757(n) = (-1)^n + a(n). (End)
a(n) = n-th forward difference of [0, 1, 1, 2, 6, 24, ...] (factorials A000142 with 0 prepended). - Michael Somos, Mar 28 2011
a(n) ~ exp(-1)*(n-1)!. - Vaclav Kotesovec, Mar 10 2014
From Vladimir Reshetnikov, Oct 29 2015: (Start)
Recurrence: a(0) = 0, a(1) = 1, a(2) = -1, a(n) = (n-3)*a(n-1) + 2*(n-2)*a(n-2) + (n-2)*a(n-3).
a(n) + a(n+1) = A000166(n). (End)
a(n) = (-1)^(n-1)*n*hypergeom([1,1,1-n], [2], 1). - Peter Luschny, May 09 2017
EXAMPLE
a(3) = 2 = 2! - 3*1! + 3*0! - 0. a(4) = 0 = 3! - 4*2! + 6*1! - 4*0! + 0. - Michael Somos, Mar 28 2011
MAPLE
a := n -> (-1)^(n-1)*n*hypergeom([1, 1, 1-n], [2], 1):
seq(simplify(a(n)), n = 0..25); # Peter Luschny, May 09 2017
MATHEMATICA
a[n_] := Sum[(-1)^k*n!/((n-k)*k!), {k, 0, n-1}]; Table[a[n], {n, 0, 19}](* Jean-François Alcover, Nov 21 2011 *)
With[{nn=30}, CoefficientList[Series[-Log[1-x] E^-x, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 28 2022 *)
PROG
(PARI) {a(n) = if( n<0, 0, sum( k=0, n-1, (-1)^k * binomial(n, k) * (n - k - 1)!))} /* Michael Somos, Jun 21 2002 */
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
EXTENSIONS
More terms from Jeffrey Shallit
More terms from Joerg Arndt, Sep 02 2013
STATUS
approved