Search: id:A001705
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%I A001705 M3944 N1625
%S A001705 0,1,5,26,154,1044,8028,69264,663696,6999840,80627040,1007441280,
%T A001705 13575738240,196287356160,3031488633600,49811492505600,867718162483200,
%U A001705 15974614352793600,309920046408806400,6320046028584960000
%N A001705 Generalized Stirling numbers: a(n) = n!*Sum[(k+1)/(n-k),{k,0,n-1}].
%C A001705 Partial sum of first n harmonic numbers multiplied by n!: a(n) = n!*Sum[Sum[1/
               k,{k,1,m}],{m,1,n}] = n!*Sum[H(m),{m,1,n}], whrere H(m) = Sum[1/k,
               {k,1,m}] = A001008(m)/A002805(m) is m-th Harmonic number.
%C A001705 In the symmetric group S_n, each permutation factors into k independent 
               cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), 
               Jun 28 2004
%C A001705 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008: 
               (Start)
%C A001705 a(n) is also the sum of the positions of the right-to-left minima in 
               all permutations of [n]. Example: a(3)=26 because the positions of 
               tle right-to-left minima in the permutations 123,132,213,231,312 
               and 321 are 123, 13, 23, 23, 3 and 3, respectively and 1+2+3+1+3+2+3+2+3+3+3=26.
%C A001705 a(n)=Sum(k*A143947(n,k),k=n..n(n+1)/2).
%C A001705 (End)
%C A001705 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 
               2009: (Start)
%C A001705 The asymptotic expansion of the higher order exponential integral E(x,
               m=2,n=2) ~ exp(-x)/x^2*(1 - 5/x + 26/x^2 - 154/x^3 + 1044/x^4 - 8028/
               x^5 + 69264/x^6 - ...) leads to the sequence given above. See A163931 
               and A028421 for more information.
%C A001705 (End)
%D A001705 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001705 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001705 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres 
               relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. 
               Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%H A001705 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=406">
               Encyclopedia of Combinatorial Structures 406</a>
%F A001705 E.g.f.: - ln ( 1 - x ) / ( 1 - x )^2. a(n) = (n+1)! * H[ n ] - n*n!, 
               H[ n ] = sum[ k=1..n ] k^-1.
%F A001705 a(n) = a(n-1)*(n+1)+n! = A000254(n+1)-A000142(n+1) = A067176(n+1, 1) 
               - Henry Bottomley (se16(AT)btinternet.com), Jan 09 2002
%F A001705 a(n)=sum((-1)^(n-1+k)*(k+1)*2^k*stirling1(n, k+1), k=0..n-1). - Borislav 
               Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
%F A001705 With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 
               0. - Ralf Stephan, Apr 16 2004
%F A001705 a(n) = n!*Sum[(k+1)/(n-k), {k, 0, n-1}] - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Oct 09 2004. Examples: a(6) = 6!*(1/6+2/5+3/4+4/3+5/2+6/1) = 8028; 
               a(20) = 20!*(1/20+2/19+3/18+4/17+5/16+...+16/5+17/4+18/3+19/2+20/
               1) = 135153868608460800000.
%F A001705 a(n) = Sum[k StirlingCycle[n+1,k+1],{k,1,n}]. - David Callan (callan(AT)stat.wisc.edu), 
               Sep 25 2006
%F A001705 For n>=1, a(n)=sum((-1)^(n-j-1)*2^j*(j+1)*stirling1(n,j+1),j=0..n-1); 
               [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]
%F A001705 a(n) = (2n+1)a(n-1) - n^2 a(n-2) [From Gary Detlefs (gdetlefs(AT)aol.com), 
               Nov 27 2009]
%e A001705 (1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...
%e A001705 Examples: a(6) = 6!*(1/6+2/5+3/4+4/3+5/2+6/1) = 8028; a(20) = 20!*(1/
               20+2/19+3/18+4/17+5/16+...+16/5+17/4+18/3+19/2+20/1) = 135153868608460800000
%p A001705 a:=n->sum(n!/k, k=2..n): seq(a(n), n=1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 22 2008
%t A001705 Table[n!*Sum[Sum[1/k,{k,1,m}],{m,1,n}],{n,0,20}] - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Apr 14 2006
%Y A001705 Cf. A000254, A006675.
%Y A001705 a(n)=A112486(n, 1).
%Y A001705 Cf. A001008, A002805.
%Y A001705 A143947 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008]
%Y A001705 Sequence in context: A045379 A053487 A082029 this_sequence A081047 A057793 
               A090226
%Y A001705 Adjacent sequences: A001702 A001703 A001704 this_sequence A001706 A001707 
               A001708
%K A001705 nonn,easy,new
%O A001705 0,3
%A A001705 N. J. A. Sloane (njas(AT)research.att.com).
%E A001705 More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 22 
               2002
%E A001705 Additional comments from Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Oct 09 2004

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