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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 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 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 Cf. A000142, A000166, A000757. Sequence in context: A154937 A037996 A299626 * A213322 A368703 A345048 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 April 15 11:27 EDT 2024. Contains 371681 sequences. (Running on oeis4.)