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%I M3944 N1625 #152 Jun 23 2022 07:00:50
%S 0,1,5,26,154,1044,8028,69264,663696,6999840,80627040,1007441280,
%T 13575738240,196287356160,3031488633600,49811492505600,
%U 867718162483200,15974614352793600,309920046408806400,6320046028584960000,135153868608460800000,3024476051557847040000
%N Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).
%C 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 the right-to-left minima in the permutations 123,132,213,231,312 and 321 are 123, 13, 23, 3, 23 and 3, respectively and 1 + 2 + 3 + 1 + 3 + 2 + 3 + 3 + 2 + 3 + 3 = 26. - _Emeric Deutsch_, Sep 22 2008
%C 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. - _Johannes W. Meijer_, Oct 20 2009
%C a(n) is the total number of cycles (excluding fixed points) in all permutations of [n+1]. - _Olivier Gérard_, Oct 23 2012; Dec 31 2012
%C A length n sequence is formed by randomly selecting (one-by-one) n real numbers in (0,1). a(n)/(n+1)! is the expected value of the sum of the new maximums in such a sequence. For example for n=3: If we select (in this order): 0.591996, 0.646474, 0.163659 we would add 0.591996 + 0.646474 which would be a bit above the average of a(3)/4! = 26/24. - _Geoffrey Critzer_, Oct 17 2013
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A001705/b001705.txt">Table of n, a(n) for n = 0..445</a> (terms 0 through 100 were provided by T. D. Noe)
%H J.-L. Baril and S. Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, preprint, 2016.
%H J.-L. Baril and S. Kirgizov, <a href="https://doi.org/10.1016/j.disc.2017.06.005">The pure descent statistic on permutations</a>, Discrete Mathematics, 340(10) (2017), 2550-2558.
%H Jean-Luc Baril and Sergey Kirgizov, <a href="https://arxiv.org/abs/2101.01928">Transformation à la Foata for special kinds of descents and excedances</a>, arXiv:2101.01928 [math.CO], 2021.
%H Gang Chen, Henrik Johansson, Fei Teng, and Tianheng Wang, <a href="https://arxiv.org/abs/2104.12726">Next-to-MHV Yang-Mills kinematic algebra</a>, arXiv:2104.12726 [hep-th], 2021. See p. 46.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=406">Encyclopedia of Combinatorial Structures 406</a>.
%H D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]
%H J. Riordan, <a href="/A000254/a000254.pdf">Letter of 04/11/74</a>.
%H J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>.
%F Partial sum of first n harmonic numbers multiplied by n!.
%F a(n) = n!*Sum_{m=1..n} Sum_{k=1..m} 1/k = n!*Sum_{m=1..n} H(m), where H(m) = Sum_{k=1..m} 1/k = A001008(m)/A002805(m) is m-th Harmonic number.
%F E.g.f.: - log (1 - x) / (1 - x)^2.
%F a(n) = (n+1)! * H(n) - n*n!, H(n) = Sum_{k=1..n} (1/k).
%F a(n) = A112486(n, 1).
%F a(n) = a(n-1)*(n+1) + n! = A000254(n+1) - A000142(n+1) = A067176(n+1, 1). - _Henry Bottomley_, Jan 09 2002
%F a(n) = Sum_{k=0..n-1} ((-1)^(n-1+k) * (k+1) * 2^k * Stirling1(n, k+1)). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
%F With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - _Ralf Stephan_, Apr 16 2004
%F a(n) = Sum_{k=1..n} (k*StirlingCycle(n+1,k+1)). - _David Callan_, Sep 25 2006
%F a(n) = Sum_{k=n..n*(n+1)/2} k*A143947(n,k). - _Emeric Deutsch_, Sep 22 2008
%F For n >= 1, a(n) = Sum_{j=0..n-1} ((-1)^(n-j-1) * 2^j * (j+1) * Stirling1(n,j+1)). - _Milan Janjic_, Dec 14 2008
%F a(n) = (2*n+1)*a(n-1) - n^2*a(n-2). - _Gary Detlefs_, Nov 27 2009
%F a(n) = (n+1)!*(H(n+1) - 1) where H(n) is the n-th harmonic number. - _Gary Detlefs_, Dec 18 2009
%F a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n+1,k+1)/k. - _Vladimir Kruchinin_, Oct 10 2016
%F a(n) = (n+1)!*Sum_{k = 1..n} (-1)^(k+1)*binomial(n+1,k+1)*k/(k+1). - _Peter Bala_, Feb 15 2022
%F a(n) = Gamma(n + 2) * (Digamma(n + 2) + EulerGamma - 1). - _Peter Luschny_, Feb 19 2022
%F From _Mélika Tebni_, Jun 22 2022: (Start)
%F a(n) = -Sum_{k=0..n} k!*A066667(n, k+1).
%F a(n) = Sum_{k=0..n} k!*A132159(n, k+1). (End)
%F a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], 1)/2. - _Peter Luschny_, Jun 22 2022
%e (1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...
%e 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. - _Alexander Adamchuk_, Oct 09 2004
%e From _Olivier Gérard_, Dec 31 2012: (Start)
%e The cycle decomposition of all permutations of 4 elements gives the following list: {{{1},{2},{3},{4}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4,3}}, {{1},{2,3,4}}, {{1},{2,4},{3}}, {{1,2},{3},{4}}, {{1,2},{3,4}}, {{1,3,2},{4}},{{1,4,3,2}}, {{1,3,4,2}}, {{1,4,2},{3}}, {{1,2,3},{4}}, {{1,2,4,3}},{{1,3},{2},{4}}, {{1,4,3},{2}}, {{1,3},{2,4}}, {{1,4,2,3}}, {{1,2,3,4}}, {{1,2,4},{3}}, {{1,3,4},{2}}, {{1,4},{2},{3}}, {{1,3,2,4}}, {{1,4},{2,3}}}.
%e Deleting the fixed points gives the following 26 items: {{3,4}, {2,3}, {2,4,3}, {2,3,4}, {2,4}, {1,2}, {1,2}, {3,4}, {1,3,2}, {1,4,3,2}, {1,3,4,2}, {1,4,2}, {1,2,3}, {1,2,4,3}, {1,3}, {1,4,3}, {1,3}, {2,4}, {1,4,2,3}, {1,2,3,4}, {1,2,4}, {1,3,4}, {1,4}, {1,3,2,4}, {1,4}, {2,3}}. (End)
%p a := n-> add((n+1)!/k, k=2..n+1): seq(a(n), n=0..21); # _Zerinvary Lajos_, Jan 22 2008; edited _Johannes W. Meijer_, Nov 28 2012
%p a := n -> ((n+1)!*(h(n+1)-1)): h := n-> harmonic(n): seq(a(n), n=0..21); # _Gary Detlefs_, Dec 18 2009; corrected by _Johannes W. Meijer_, Nov 28 2012
%t Table[n!*Sum[Sum[1/k,{k,1,m}], {m,1,n}], {n,0,20}] (* _Alexander Adamchuk_, Apr 14 2006 *)
%t a[n_] := (n + 1)! (EulerGamma - 1 + PolyGamma[n + 2]);
%t Table[a[n], {n, 0, 21}] (* _Peter Luschny_, Feb 19 2022 *)
%o (Maxima)
%o a(n):=n!*sum(((-1)^(k+1)*binomial(n+1,k+1))/k,k,1,n); /* _Vladimir Kruchinin_, Oct 10 2016 */
%o (PARI) for(n=0,25, print1(n!*sum(k=0,n-1,(k+1)/(n-k)), ", ")) \\ _G. C. Greubel_, Jan 20 2017
%o (Python)
%o from math import factorial
%o def A001705(n):
%o f = factorial(n)
%o return sum(f*(k+1)//(n-k) for k in range(n)) # _Chai Wah Wu_, Jun 23 2022
%Y Cf. A000254 (total number of cycles in permutations, including fixed points).
%Y Cf. A002104 (number of different cycles in permutations, without fixed points).
%Y Cf. A006231 (number of different cycles in permutations, including fixed points).
%Y Cf. A001008, A002805, A006675, A066667, A132159, A143947.
%Y Related to n!*the k-th successive summation of the harmonic numbers:
%Y (k=0) A000254, (k=1) A001705, (k=2) A001711, (k=3) A001716,
%Y (k=4) A001721, (k=5) A051524, (k=6) A051545, (k=7) A051560,
%Y (k=8) A051562, (k=9) A051564.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Sascha Kurz_, Mar 22 2002
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