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A001705
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Generalized Stirling numbers: a(n) = n!*Sum[(k+1)/(n-k),{k,0,n-1}].
(Formerly M3944 N1625)
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39
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0, 1, 5, 26, 154, 1044, 8028, 69264, 663696, 6999840, 80627040, 1007441280, 13575738240, 196287356160, 3031488633600, 49811492505600, 867718162483200, 15974614352793600, 309920046408806400, 6320046028584960000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
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
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008: (Start)
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.
a(n)=Sum(k*A143947(n,k),k=n..n(n+1)/2).
(End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)
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.
(End)
a(n)= (n+1)!*(h(n+1)-1) where h(n) is the n'th harmonic number [From Gary Detlefs (gdetlefs(AT)aol.com), Dec 18 2009]
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REFERENCES
| 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.
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).
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 406
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FORMULA
| 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.
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
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
With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004
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.
a(n) = Sum[k StirlingCycle[n+1,k+1],{k,1,n}]. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
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]
a(n) = (2n+1)a(n-1) - n^2 a(n-2) [From Gary Detlefs (gdetlefs(AT)aol.com), Nov 27 2009]
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EXAMPLE
| (1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...
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
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MAPLE
| a:=n->sum(n!/k, k=2..n): seq(a(n), n=1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 22 2008
h:= n-> convert(evalf(harmonic(n)), fraction): for i from 1 to 30 do print((i+1)!*(h(i+1)-1)) od; [From Gary Detlefs (gdetlefs(AT)aol.com), Dec 18 2009]
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MATHEMATICA
| Table[n!*Sum[Sum[1/k, {k, 1, m}], {m, 1, n}], {n, 0, 20}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 14 2006
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CROSSREFS
| Cf. A000254, A006675.
a(n)=A112486(n, 1).
Cf. A001008, A002805.
A143947 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008]
Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564 [From Gary Detlefs (gdetlefs(AT)aol.com) Jan 04 2011]
Sequence in context: A183161 A082029 * A081047 A057793 A090226 A094422
Adjacent sequences: A001702 A001703 A001704 * A001706 A001707 A001708
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 22 2002
Additional comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 09 2004
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