A002741 Logarithmic numbers: expansion of the e.g.f. -log(1-x) * e^(-x).

0, 1, -1, 2, 0, 9, 35, 230, 1624, 13209, 120287, 1214674, 13469896, 162744945, 2128047987, 29943053062, 451123462672, 7245940789073, 123604151490591, 2231697509543362, 42519034050101744, 852495597142800377, 17942811657908144163, 395553947953212635718, 9114871523102565301544, 219135339782236105192745

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Examples

			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
		

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]
• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A000142, A000166, A000757.

Programs

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 *)

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 */

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

Extensions

More terms from Jeffrey Shallit

More terms from Joerg Arndt, Sep 02 2013