A002104 Logarithmic numbers.
0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, 60683990530225, 1027542662934915, 18430998766219336, 349096664728623336, 6962409983976703337, 145841989688186383359, 3201192743180799343844
Offset: 0
Examples
From _Reinhard Zumkeller_, Oct 26 2010: (Start)
a(3) = #{[1], [1,2], [1,2,3], [1,3], [1,3,2], [2], [2,3], [3]} = 8;
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)
G.f. = x + 3*x^2 + 8*x^3 + 24*x^4 + 89*x^5 + 415*x^6 + 2372*x^7 + ...
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 (corrected by Michel Marcus, Jan 19 2019)
- J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
- Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, and Jordi Tura, Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes, arXiv:2407.08783 [quant-ph], 2024. See p. 9.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 116
- J. C. Tiernan, An efficient search algorithm to find the elementary circuits of a graph, Commun. ACM, 13 (1970), 722-726.
- Index entries for sequences related to logarithmic numbers
Programs
-
Haskell
import Data.List (subsequences, permutations) a002104 = length . filter (\xs -> head xs == minimum xs) . tail . choices . enumFromTo 1 where choices = concat . map permutations . subsequences -- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010 -
Maple
a := proc(n) option remember; ifelse(n < 2, n, n*a(n-1) - (n-1)*a(n-2) + 1) end: seq(a(n), n = 0..23); # Peter Luschny, Dec 05 2023
-
Mathematica
Table[Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}],{n,1,30}] (* Alexander Adamchuk, Jul 05 2006 *) a[n_] = n*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]); Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 29 2011 *) -
PARI
x='x+O('x^99); concat([0], Vec(serlaplace(-log(1-x)*exp(x)))) \\ Altug Alkan, Dec 17 2017 -
PARI
{a(n) = sum(k=0, n-1, binomial(n, k) * (n-k-1)!)}; /* Michael Somos, May 08 2019 */
Formula
E.g.f.: -log(1 - x) * exp(x).
a(n) = Sum_{k=1..n} Sum_{i=0..n-k} (n-k)!/i!.
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
a(n+1) - a(n) = A000522(n).
a(n) = sum{k=0..n-1, binomial(n, k)*(n-k-1)!}, row sums of A111492. - Paul Barry, Aug 26 2004
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
For n > 1, the arithmetic mean of the first n terms is a(n-1) + 1. - Franklin T. Adams-Watters, May 20 2010
a(n) = n * 3F1((1,1,1-n); (2); -1). - Jean-François Alcover, Mar 29 2011
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
From Emanuele Munarini, Dec 16 2017: (Start)
The generating series A(x) = -exp(x)*log(1-x) satisfies the differential equations:
(1-x)*A'(x) - (1-x)*A(x) = exp(x)
(1-x)*A''(x) - (3-2*x)*A'(x) + (2-x)*A(x) = 0.
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)
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
a(n) ~ exp(1)*(n-1)!. - Vaclav Kotesovec, Mar 10 2014
a(n) = n*a(n-1) - (n-1)*a(n-2) + 1, a(0) = 0, a(1) = 1. - Richard R. Forberg, Dec 15 2014
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
From Peter Bala, Sep 12 2022: (Start)
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, ...].
Conjectures:
1) for n, m >= 0, k >= 2, a(n + m*2^k) - a(n) is divisible by 2^k.
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)
a(n) = Integral_{t=0..oo} ((t + 1)^n - 1)/(t*e^t) dt. - Velin Yanev, Apr 13 2024
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
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
Comments