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A001705 Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).
(Formerly M3944 N1625)
48
0, 1, 5, 26, 154, 1044, 8028, 69264, 663696, 6999840, 80627040, 1007441280, 13575738240, 196287356160, 3031488633600, 49811492505600, 867718162483200, 15974614352793600, 309920046408806400, 6320046028584960000, 135153868608460800000, 3024476051557847040000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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

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

REFERENCES

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

G. C. Greubel, Table of n, a(n) for n = 0..445 (terms 0 through 100 were provided by T. D. Noe)

J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.

J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, 340(10) (2017), 2550-2558.

Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021.

Gang Chen, Henrik Johansson, Fei Teng, and Tianheng Wang, Next-to-MHV Yang-Mills kinematic algebra, arXiv:2104.12726 [hep-th], 2021. See p. 46.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 406.

D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]

J. Riordan, Letter of 04/11/74.

J. Riordan, Letter, Jul 06 1978.

FORMULA

Partial sum of first n harmonic numbers multiplied by n!.

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.

E.g.f.: - log (1 - x) / (1 - x)^2.

a(n) = (n+1)! * H(n) - n*n!, H(n) = Sum_{k=1..n} (1/k).

a(n) = A112486(n, 1).

a(n) = a(n-1)*(n+1) + n! = A000254(n+1) - A000142(n+1) = A067176(n+1, 1). - Henry Bottomley, Jan 09 2002

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

With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004

a(n) = Sum_{k=1..n} (k*StirlingCycle(n+1,k+1)). - David Callan, Sep 25 2006

a(n) = Sum_{k=n..n*(n+1)/2} k*A143947(n,k). - Emeric Deutsch, Sep 22 2008

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

a(n) = (2*n+1)*a(n-1) - n^2*a(n-2). - Gary Detlefs, Nov 27 2009

a(n) = (n+1)!*(H(n+1) - 1) where H(n) is the n-th harmonic number. - Gary Detlefs, Dec 18 2009

a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n+1,k+1)/k. - Vladimir Kruchinin, Oct 10 2016

a(n) = (n+1)!*Sum_{k = 1..n} (-1)^(k+1)*binomial(n+1,k+1)*k/(k+1). - Peter Bala, Feb 15 2022

a(n) = Gamma(n + 2) * (Digamma(n + 2) + EulerGamma - 1). - Peter Luschny, Feb 19 2022

From Mélika Tebni, Jun 22 2022: (Start)

a(n) = -Sum_{k=0..n} k!*A066667(n, k+1).

a(n) = Sum_{k=0..n} k!*A132159(n, k+1). (End)

a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], 1)/2. - Peter Luschny, Jun 22 2022

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. - Alexander Adamchuk, Oct 09 2004

From Olivier Gérard, Dec 31 2012: (Start)

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}}}.

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)

MAPLE

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

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

MATHEMATICA

Table[n!*Sum[Sum[1/k, {k, 1, m}], {m, 1, n}], {n, 0, 20}] (* Alexander Adamchuk, Apr 14 2006 *)

a[n_] := (n + 1)! (EulerGamma - 1 + PolyGamma[n + 2]);

Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 19 2022 *)

PROG

(Maxima)

a(n):=n!*sum(((-1)^(k+1)*binomial(n+1, k+1))/k, k, 1, n); /* Vladimir Kruchinin, Oct 10 2016 */

(PARI) for(n=0, 25, print1(n!*sum(k=0, n-1, (k+1)/(n-k)), ", ")) \\ G. C. Greubel, Jan 20 2017

(Python)

from math import factorial

def A001705(n):

    f = factorial(n)

    return sum(f*(k+1)//(n-k) for k in range(n)) # Chai Wah Wu, Jun 23 2022

CROSSREFS

Cf. A000254 (total number of cycles in permutations, including fixed points).

Cf. A002104 (number of different cycles in permutations, without fixed points).

Cf. A006231 (number of different cycles in permutations, including fixed points).

Cf. A001008, A002805, A006675, A066667, A132159, A143947.

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.

Sequence in context: A263134 A082029 A081047 * A185108 A302442 A349882

Adjacent sequences:  A001702 A001703 A001704 * A001706 A001707 A001708

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Sascha Kurz, Mar 22 2002

STATUS

approved

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Last modified July 7 02:44 EDT 2022. Contains 355141 sequences. (Running on oeis4.)