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A002741 Logarithmic numbers: expansion of -log(1-x) e^(-x).
(Formerly M0037 N0010)
27
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

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

FORMULA

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). - Michael Somos, Jun 21 2002

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-Fran├žois Alcover, Nov 21 2011 *)

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

Cf. A000142, A000166, A000757.

Sequence in context: A154937 A037996 A299626 * A213322 A151887 A303350

Adjacent sequences:  A002738 A002739 A002740 * A002742 A002743 A002744

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jeffrey Shallit

More terms from Joerg Arndt, Sep 02 2013

STATUS

approved

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Last modified December 11 07:38 EST 2019. Contains 329914 sequences. (Running on oeis4.)