%I M2749 N1105 #121 Jul 20 2024 10:54:29

%S 0,1,3,8,24,89,415,2372,16072,125673,1112083,10976184,119481296,

%T 1421542641,18348340127,255323504932,3809950977008,60683990530225,

%U 1027542662934915,18430998766219336,349096664728623336,6962409983976703337,145841989688186383359,3201192743180799343844

%N Logarithmic numbers.

%C Prime p divides a(p+1). - _Alexander Adamchuk_, Jul 05 2006

%C Also number of lists of elements from {1,..,n} with (1st element) = (smallest element), where a list means an ordered subset (cf. A000262), see also Haskell program. - _Reinhard Zumkeller_, Oct 26 2010

%C a(n+1) = p_n(-1) where p_n(x) is the unique degree-n polynomial such that p_n(k) = A133942(k) for k = 0, 1, ..., n. - _Michael Somos_, Apr 30 2012

%C a(n) = A006231(n) + n. - _Geoffrey Critzer_, Oct 04 2012

%D J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002104/b002104.txt">Table of n, a(n) for n = 0..100</a> (corrected by Michel Marcus, Jan 19 2019)

%H J. M. Gandhi, <a href="/A002741/a002741.pdf">On logarithmic numbers</a>, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]

%H Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, and Jordi Tura, <a href="https://arxiv.org/abs/2407.08783">Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes</a>, arXiv:2407.08783 [quant-ph], 2024. See p. 9.

%H INRIA Algorithms Project, <a href="https://ecs.inria.fr/services/structure?nbr=116">Encyclopedia of Combinatorial Structures 116</a>

%H J. C. Tiernan, <a href="https://dx.doi.org/10.1145/362814.362819">An efficient search algorithm to find the elementary circuits of a graph</a>, Commun. ACM, 13 (1970), 722-726.

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F E.g.f.: -log(1 - x) * exp(x).

%F a(n) = Sum_{k=1..n} Sum_{i=0..n-k} (n-k)!/i!.

%F a(n) = Sum_{k=1..n} n(n-1)...(n-k+1)/k = A006231(n) + n - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001

%F a(n+1) - a(n) = A000522(n).

%F a(n) = sum{k=0..n-1, binomial(n, k)*(n-k-1)!}, row sums of A111492. - _Paul Barry_, Aug 26 2004

%F a(n) = Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}]. a(n) = Sum[A000522(m),{m,0,n-1}]. - _Alexander Adamchuk_, Jul 05 2006

%F For n > 1, the arithmetic mean of the first n terms is a(n-1) + 1. - _Franklin T. Adams-Watters_, May 20 2010

%F a(n) = n * 3F1((1,1,1-n); (2); -1). - _Jean-François Alcover_, Mar 29 2011

%F Conjecture: a(n) +(-n-1)*a(n-1) +2*(n-1)*a(n-2) +(-n+2)*a(n-3)=0. - _R. J. Mathar_, Dec 02 2012

%F From _Emanuele Munarini_, Dec 16 2017: (Start)

%F The generating series A(x) = -exp(x)*log(1-x) satisfies the differential equations:

%F (1-x)*A'(x) - (1-x)*A(x) = exp(x)

%F (1-x)*A''(x) - (3-2*x)*A'(x) + (2-x)*A(x) = 0.

%F From the first one, we have the recurrence reported below by R. R. Forberg. From the second one, we have the recurrence conjectured above. (End)

%F G.f.: conjecture: T(0)*x/(1-2*x)/(1-x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 18 2013

%F a(n) ~ exp(1)*(n-1)!. - _Vaclav Kotesovec_, Mar 10 2014

%F a(n) = n*a(n-1) - (n-1)*a(n-2) + 1, a(0) = 0, a(1) = 1. - _Richard R. Forberg_, Dec 15 2014

%F a(n) = A007526(n) + A006231(n+1) - A030297(n). - _Anton Zakharov_, Sep 05 2016

%F 0 = +a(n)*(+a(n+1) -4*a(n+2) +4*a(n+3) -a(n+4)) +a(n+1)*(+2*a(n+2) -5*a(n+3) +2*a(n+4)) +a(n+2)*(+2*a(n+2) -a(n+3) -a(n+4)) +a(n+3)*(+a(n+3)) for all n>=0. - _Michael Somos_, May 08 2019

%F From _Peter Bala_, Sep 12 2022: (Start)

%F For n, m >= 0, a(n) - a(n + m) == ( a(1) - a(m) ) (mod m). The sequence {mod(a(1) - a(m+1), m): m >= 1} begins [0, 1, 1, 0, 1, 5, 1, 0, 3, 7, 1, 4, 1, 9, 8, 0, 1, 15, 1, 4, ...].

%F Conjectures:

%F 1) for n, m >= 0, k >= 2, a(n + m*2^k) - a(n) is divisible by 2^k.

%F 2) for n >= 0, a(n + m*p^k) - a(n) + m*p^(k-1) is divisible by p^k for all positive integers m and k, and for all odd primes p. The particular case n = m = k = 1 is stated in the Comments section by Adamchuk. (End)

%F a(n) = Integral_{t=0..oo} ((t + 1)^n - 1)/(t*e^t) dt. - _Velin Yanev_, Apr 13 2024

%F a(n) = Gamma(n)*(e - ((-1)^n)*Gamma(1 - n, -1)) + hypergeom([1, 1], [2, n + 2], 1)/(n + 1) - polygamma(n) - 1/n + i*Pi for n > 0, where polygamma is the digamma function and the bivariate gamma function is the upper incomplete gamma function. - _Velin Yanev_, Apr 13 2024

%e From _Reinhard Zumkeller_, Oct 26 2010: (Start)

%e a(3) = #{[1], [1,2], [1,2,3], [1,3], [1,3,2], [2], [2,3], [3]} = 8;

%e a(4) = #{[1], [1,2], [1,2,3], [1,2,3,4], [1,2,4], [1,2,4,3], [1,3], [1,3,2], [1,3,2,4], [1,3,4], [1,3,4,2], [1,4], [1,4,2], [1,4,2,3], [1,4,3], [1,4,3,2], [2], [2,3], [2,3,4], [2,4], [2,4,3], [3], [3,4], [4]} = 24. (End)

%e G.f. = x + 3*x^2 + 8*x^3 + 24*x^4 + 89*x^5 + 415*x^6 + 2372*x^7 + ...

%p a := proc(n) option remember; ifelse(n < 2, n, n*a(n-1) - (n-1)*a(n-2) + 1) end:

%p seq(a(n), n = 0..23); # _Peter Luschny_, Dec 05 2023

%t Table[Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}],{n,1,30}] (* _Alexander Adamchuk_, Jul 05 2006 *)

%t a[n_] = n*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]); Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Mar 29 2011 *)

%o (Haskell)

%o import Data.List (subsequences, permutations)

%o a002104 = length . filter (\xs -> head xs == minimum xs) .

%o tail . choices . enumFromTo 1

%o where choices = concat . map permutations . subsequences

%o -- _Reinhard Zumkeller_, Feb 21 2012, Oct 25 2010

%o (PARI) x='x+O('x^99); concat([0], Vec(serlaplace(-log(1-x)*exp(x)))) \\ _Altug Alkan_, Dec 17 2017

%o (PARI) {a(n) = sum(k=0, n-1, binomial(n, k) * (n-k-1)!)}; /* _Michael Somos_, May 08 2019 */

%Y Cf. A001338, A006231, A007526, A030297, A133942.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001