OFFSET
0,3
COMMENTS
Prime p divides a(p+1). - Alexander Adamchuk, Jul 05 2006
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
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
a(n) = A006231(n) + n. - Geoffrey Critzer, Oct 04 2012
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.
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
EXAMPLE
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 + ...
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 *)
PROG
(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
(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 */
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
STATUS
approved