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A002741 Logarithmic numbers: expansion of the e.g.f. -log(1-x) * e^(-x).
(Formerly M0037 N0010)
37
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A154937 A037996 A299626 * A213322 A368703 A345048
KEYWORD
sign,easy,nice
AUTHOR
EXTENSIONS
More terms from Jeffrey Shallit
More terms from Joerg Arndt, Sep 02 2013
STATUS
approved

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Last modified April 15 11:27 EDT 2024. Contains 371681 sequences. (Running on oeis4.)