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A002104
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Logarithmic numbers.
(Formerly M2749 N1105)
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10
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0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, 60683990530225, 1027542662934915, 18430998766219336, 349096664728623336
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Prime p divides a(p+1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
Also number of lists of {1,..,n} with (1st element) = (smallest element), where a list means an ordered subset (cf. A000262), see also Haskell program. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 26 2010]
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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).
J. C. Tiernan, An efficient search algorithm to find the elementary circuits of a graph, Commun. ACM, 13 (1970), 722-726.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 116
Index entries for sequences related to logarithmic numbers
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FORMULA
| E.g.f.: -ln(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)!} - Paul Barry (pbarry(AT)wit.ie), 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 (alex(AT)kolmogorov.com), Jul 05 2006
For n > 1, the arithmetic mean of the first n terms is a(n-1) + 1. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 20 2010]
a(n) = n * 3F1((1,1,1-n); (2); -1) [From Jean-François Alcover, Mar 29 2011]
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EXAMPLE
| Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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)
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MATHEMATICA
| Table[Sum[Sum[m!/k!, {k, 0, m}], {m, 0, n-1}], {n, 1, 30}] (* Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006 *)
a[n_] = n*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]); Table[a[n], {n, 1, 20}] (* From Jean-François Alcover, Mar 29 2011 *)
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PROG
| Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 26 2010: (Start)
(Haskell)
import Data.List (subsequences, permutations)
choices xs = concat $ map permutations $ subsequences xs
a002104 n = length $ filter (\xs -> head xs == minimum xs) $
........................... tail $ choices [1..n]
-- eop. (End)
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CROSSREFS
| Cf. A006231.
Cf. A001338.
Sequence in context: A134165 A071016 A174662 * A102919 A102476 A180380
Adjacent sequences: A002101 A002102 A002103 * A002105 A002106 A002107
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
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