A159324 -id:A159324 - cp's OEIS frontend

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000

Offset: 0

N. J. A. Sloane

Number of permutations of n+1 elements with exactly two cycles.
Number of cycles in all permutations of [n]. Example: a(3) = 11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles altogether. - Emeric Deutsch, Aug 12 2004
Row sums of A094310: 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
The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch, Aug 12 2006
a(n) is divisible by n for all composite n >= 6. a(2*n) is divisible by 2*n + 1. - Leroy Quet, May 20 2007
For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g., for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec, Jan 13 2008, Mar 26 2008
The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence. (1 + 1/2 = 2/2 + 1/2 = 3/2; 3/2 + 1/3 = 9/6 + 2/6 = 11/6; 11/6 + 1/4 = 44/24 + 6/24 = 50/24;...). The denominator of this fraction is n!*A000142. - Eric Desbiaux, Jan 07 2009
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/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 number of permutations of [n+1] containing exactly 2 cycles. Example: a(2) = 3 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3] with exactly 2 cycles. - Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009
It appears that, with the exception of n= 4, a(n) mod n = 0 if n is composite and = n-1 if n is prime. - Gary Detlefs, Sep 11 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012
Numerator of harmonic number H(n) = Sum_{i=1..n} 1/i when not reduced. See A001008 (Wolstenholme numbers) for the reduced numerators. - Rahul Jha, Feb 18 2015
The Stirling transform of this sequence is A222058(n) (Harmonic-geometric numbers). - Anton Zakharov, Aug 07 2016
a(n) is the (n-1)-st elementary symmetric function of the first n numbers. - Anton Zakharov, Nov 02 2016
The n-th iterated integral of log(x) is x^n * (n! * log(x) - a(n))/(n!)^2 + a polynomial of degree n-1 with arbitrary coefficients. This can be proven using the recurrence relation a(n) = (n-1)! + n*a(n-1). - Mohsen Maesumi, Oct 31 2018
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019
Total number of left-to-right maxima (or minima) in all permutations of [n]. a(3) = 11 = 3+2+2+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21. - Alois P. Heinz, Aug 01 2020

			(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
Shanzhen Gao, Permutations with Restricted Structure (in preparation).
K. Javorszky, Natural Orders: De Ordinibus Naturalibus, 2016, ISBN 978-3-99057-139-2.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..449 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159 (1996), 29-42.
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.
FindStat - Combinatorial Statistic Finder, The number of cycles in the cycle decomposition of a permutation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 31.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
J. Riordan, Letter of 04/11/74.
John A. Rochowicz Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE), 8(2) (2015), Article 4.
N. A. Rosenberg, Informativeness of genetic markers for inference of ancestry, American Journal of Human Genetics 73 (2003), 1402-1422.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), #10.6.7, Section 4.3.2.
J. Scholes, 53rd Putnam 1992, Problem B5.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.

Cf. A000399, A000454, A000482, A001233, A001234, A243569, A243570.
Cf. A000774, A004041, A024167, A046674, A049034, A008275.
Cf. A021009, A081358, A092691, A094587, A151881, A121633.
With signs: A081048.
Column 1 in triangle A008969.
Row sums of A136662.

a:=[]; for n in [1..22] do a:=a cat [Abs(StirlingFirst(n,2))]; end for; a; // Marius A. Burtea, Jan 01 2020

A000254 := proc(n) option remember; if n<=1 then n else n*A000254(n-1)+(n-1)!; fi; end: seq(A000254(n),n=0..21);
a := n -> add(n!/k, k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008

Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] (* Wouter Meeussen *)
Table[ n!*HarmonicNumber[n], {n, 0, 19}] (* Robert G. Wilson v, May 21 2005 *)
Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Abs[StirlingS1[Range[20],2]] (* Harvey P. Dale, Aug 16 2011 *)
Table[Gamma'[n + 1] /. EulerGamma -> 0, {n, 0, 30}] (* Li Han, Feb 14 2024*)

a(n):=(-1)^(n+1)/2*(n+1)*sum(k*bern(k-1)*stirling1(n,k),k,1,n); /* Vladimir Kruchinin, Nov 20 2016 */

A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:

{a(n) = if( n<0, 0, (n+1)! / 2 * sum( k=1, n, 1 / k / (n+1-k)))} /* Michael Somos, Feb 05 2004 */

[stirling_number1(i, 2) for i in range(1, 22)]  # Zerinvary Lajos, Jun 27 2008

Let P(n,X) = (X+1)*(X+2)*(X+3)*...*(X+n); then a(n) is the coefficient of X; or a(n) = P'(n,0). - Benoit Cloitre, May 09 2002
Sum_{k > 0} a(k) * x^k/ k!^2 = exp(x) *(Sum_{k>0} (-1)^(k+1) * x^k / (k * k!)). - Michael Somos, Mar 24 2004; corrected by Warren D. Smith, Feb 12 2006
a(n) is the coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
a(n) = n! * Sum_{i=1..n} 1/i = n! * H(n), where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.
a(n) ~ 2^(1/2)*Pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: log(1 - x) / (x-1). (= (log(1 - x))^2 / 2 if offset 1). - Michael Somos, Feb 05 2004
D-finite with recurrence: a(n) = a(n-1) * (2*n - 1) - a(n-2) * (n - 1)^2, if n > 1. - Michael Somos, Mar 24 2004
a(n) = A081358(n)+A092691(n). - Emeric Deutsch, Aug 12 2004
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic, Jan 29 2005
p^2 divides a(p-1) for prime p > 3. a(n) = (Sum_{i=1..n} 1/i) / Product_{i=1..n} 1/i. - Alexander Adamchuk, Jul 11 2006
a(n) = 3* A001710(n) + 2* A001711(n-3) for n > 2; e.g., 11 = 3*3 + 2*1, 50 = 3*12 + 2*7, 274 = 3*60 + 2*47, ... - Gary Detlefs, May 24 2010
a(n) = A138772(n+1) - A159324(n). - Gary Detlefs, Jul 05 2010
a(n) = A121633(n) + A002672(n). - Gary Detlefs, Jul 18 2010
a(n+1) = Sum_{i=1..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 14 2010
From Gary Detlefs, Sep 11 2010: (Start)
a(n) = (a(n-1)*(n^2 - 2*n + 1) + (n + 1)!)/(n - 1) for n > 2.
It appears that, with the exception of n = 2, (a(n+1)^2 - a(n)^2) mod n^2 = 0 if n is composite and 4*n if n is prime.
It appears that, with the exception of n = 2, (a(n+1)^3 - a(n)^2) mod n = 0 if n is composite and n - 2 if n is prime.
It appears that, with the exception of n = 2, (a(n)^2 + a(n+1)^2) mod n = 0 if n is composite and = 2 if n is prime. (End)
a(n) = Integral_{x=0..oo} (x^n - n!)*log(x)*exp(-x) dx. - Groux Roland, Mar 28 2011
a(n) = 3*n!/2 + 2*(n-2)!*Sum_{k=0..n-3} binomial(k+2,2)/(n-2-k) for n >= 2. - Gary Detlefs, Sep 02 2011
a(n)/(n-1)! = ml(n) = n*ml(n-1)/(n-1) + 1 for n > 1, where ml(n) is the average number of random draws from an n-set with replacement until the total set has been observed. G.f. of ml: x*(1 - log(1 - x))/(1 - x)^2. - Paul Weisenhorn, Nov 18 2011
a(n) = det(|S(i+2, j+1)|, 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
E.g.f.: x/(1 - x)*E(0)/2, where E(k) = 2 + E(k+1)*x*(k + 1)/(k + 2). - Sergei N. Gladkovskii, Jun 01 2013 [Edited by Michael Somos, Nov 28 2013]
0 = a(n) * (a(n+4) - 6*a(n+3) + 7*a(n+2) - a(n+1)) - a(n+1) * (4*a(n+3) - 6*a(n+2) + a(n+1)) + 3*a(n+2)^2 unless n=0. - Michael Somos, Nov 28 2013
For a simple way to calculate the sequence, multiply n! by the integral from 0 to 1 of (1 - x^n)/(1 - x) dx. - Rahul Jha, Feb 18 2015
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Inverse binomial transform of A073596.
a(n) ~ sqrt(2*Pi*n) * n^n * (log(n) + gamma)/exp(n), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = ((-1)^(n+1)/2*(n+1))*Sum_{k=1..n} k*Bernoulli(k-1)*Stirling1(n,k). - Vladimir Kruchinin, Nov 20 2016
a(n) = (n)! * (digamma(n+1) + gamma), where gamma is the Euler-Mascheroni constant A001620. - Pedro Caceres, Mar 10 2018
From Andy Nicol, Oct 21 2021: (Start)
Gamma'(x) = a(x-1) - (x-1)!*gamma, where Gamma'(x) is the derivative of the gamma function at positive integers and gamma is the Euler-Mascheroni constant. E.g.:
Gamma'(1) = -gamma, Gamma'(2) = 1-gamma, Gamma'(3) = 3-2*gamma,
Gamma'(22) = 186244810780170240000 - 51090942171709440000*gamma. (End)
From Peter Bala, Feb 03 2022: (Start)
The following are all conjectural:
E.g.f.: for nonzero m, (1/m)*Sum_{n >= 1} (-1)^(n+1)*(1/n)*binomial(m*n,n)* x^n/(1 - x)^(m*n+1) = x + 3*x^2/2! + 11*x^3/3! + 50*x^4/4! + ....
For nonzero m, a(n) = (1/m)*n!*Sum_{k = 1..n} (-1)^(k+1)*(1/k)*binomial(m*k,k)* binomial(n+(m-1)*k,n-k).
a(n)^2 = (1/2)*n!^2*Sum_{k = 1..n} (-1)^(k+1)*(1/k^2)*binomial(n,k)* binomial(n+k,k). (End)
From Mélika Tebni, Jun 20 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A021009(n, k+1).
a(n) = Sum_{k=0..n} k!*A094587(n, k+1). (End)
a(n) = n! * 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n - 1)^2/((2*n - 1)))))). - Peter Bala, Mar 16 2024

A288964 Number of key comparisons to sort all n! permutations of n elements by quicksort.

Original entry on oeis.org

0, 0, 2, 16, 116, 888, 7416, 67968, 682272, 7467840, 88678080, 1136712960, 15655438080, 230672171520, 3621985113600, 60392753971200, 1065907048550400, 19855856150323200, 389354639411404800, 8017578241634304000, 172991656889856000000, 3903132531903897600000

Offset: 0

Daniel Krenn, Jun 20 2017

nonn

From Petros Hadjicostas, May 04 2020: (Start)
Depending on the assumptions used in the literature, the average number to sort n items in random order by quicksort appears as -C*n + 2*(1+n)*HarmonicNumber(n), where C = 2, 3, or 4. (To get the number of key comparisons to sort all n! permutations of n elements by quicksort, multiply this number by n!.)
For this sequence, we have C = 4. The corresponding number of key comparisons to sort all n! permutations of n elements by quicksort when C = 3 is A063090(n). Thus, a(n) = A063090(n) - n!*n.
For more details, see my comments and references for sequences A093418, A096620, and A115107. (End)

Daniel Krenn, Table of n, a(n) for n = 0..100
Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Lucky Cars and the Quicksort Algorithm, arXiv:2306.13065 [math.CO], 2023.
Index entries for sequences related to sorting

Cf. A063090, A093418, A096620, A115107, A117627, A117628, A159324, A288965.

a:= proc(n) option remember; `if`(n<3, n*(n-1),
      ((2*n^2-3*n-1)*a(n-1)-(n-1)^2*n*a(n-2))/(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 21 2017

a[n_] := n! (2(n+1)HarmonicNumber[n] - 4n);
a /@ Range[0, 25] (* Jean-François Alcover, Oct 29 2020 *)

[n.factorial() * (2*(n+1)*sum(1/k for k in srange(1, n+1)) - 4*n) for n in srange(21)]

# alternative using the recurrence relation
@cached_function
def c(n):
    if n <= 1:
        return 0
    return (n - 1) + 2/n*sum(c(i) for i in srange(n))
[n.factorial() * c(n) for n in srange(21)]

a(n) = n!*(2*(n+1)*H(n) - 4*n).
c(n) = a(n) / n! satisfies c(n) = (n-1) + 2/n * Sum_{i < n} c(i).
a(n) = 2*A002538(n-1), n >= 2. - Omar E. Pol, Jun 20 2017
E.g.f.: -2*log(1-x)/(1-x)^2 - 2*x/(1-x)^2. - Daniel Krenn, Jun 20 2017
a(n) = ((2*n^2-3*n-1)*a(n-1) -(n-1)^2*n*a(n-2))/(n-2) for n >= 3, a(n) = n*(n-1) for n < 3. - Alois P. Heinz, Jun 21 2017
From Petros Hadjicostas, May 03 2020: (Start)
a(n) = A063090(n) - n!*n for n >= 1.
a(n) = n!*A115107(n)/A096620(n) = n!*(-n + A093418(n)/A096620(n)). (End)

A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 27, 8, 1, 360, 168, 59, 12, 1, 2520, 1200, 463, 119, 17, 1, 20160, 9720, 3978, 1177, 221, 23, 1, 181440, 88200, 37566, 12217, 2724, 382, 30, 1, 1814400, 887040, 388728, 135302, 34009, 5780, 622, 38, 1, 19958400, 9797760, 4385592, 1606446, 441383, 86029, 11378, 964, 47, 1

Offset: 1

Wouter Meeussen, Dec 26 2012

Row sums are n!*n = A001563(n) (see example).

For fixed k>=1, A185105(n,k) ~ n!*n/2^k. - Vaclav Kotesovec, Apr 25 2017

			The six permutations of n=3 in ordered cycle form are:
{ {1}, {2}, {3}    }
{ {1}, {2, 3}, {}  }
{ {1, 2}, {3}, {}  }
{ {1, 2, 3}, {}, {}}
{ {1, 3, 2}, {}, {}}
{ {1, 3}, {2}, {}  }
.
The lengths of the cycles in position k=1 sum to 12, those of the cycles in position k=2 sum to 5 and those of the cycles in position k=3 sum to 1.
Triangle begins:
       1;
       3,     1;
      12,     5,     1;
      60,    27,     8,     1;
     360,   168,    59,    12,    1;
    2520,  1200,   463,   119,   17,   1;
   20160,  9720,  3978,  1177,  221,  23,  1;
  181440, 88200, 37566, 12217, 2724, 382, 30, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Permutation

Columns k=1-10 give: A001710(n+1), A138772, A159324(n-1)/2 or A285231, A285232, A285233, A285234, A285235, A285236, A285237, A285238.
T(2n,n) gives A285239.
Cf. A000142, A001563, A270236, A284816, A285439.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+coeff(p, x, 0)*j*x^i)(b(n-j, i+1))*
       binomial(n-1, j-1)*(j-1)!, j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 15 2017

Table[it = Join[RotateRight /@ ToCycles[#], Table[{}, {k}]] & /@ Permutations[Range[n]]; Tr[Length[Part[#, k]]& /@ it], {n, 7}, {k, n}]
(* Second program: *)
b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 30 2018, after Alois P. Heinz *)

More terms from Alois P. Heinz, Apr 15 2017

A288970 Number of key comparisons to sort all n! permutations of n elements by the optimal trial-pivot quicksort.

Original entry on oeis.org

0, 0, 2, 16, 112, 848, 7032, 64056, 639888, 6974928, 82531296, 1054724256, 14487894144, 212971227264

Offset: 0

Daniel Krenn, Jun 20 2017

The 3 pivot elements are chosen from fixed indices (e.g., the last 3 elements). The "optimal" strategy minimizes, after the choice of the pivots is done, the expected partitioning cost.

M. Aumüller and M. Dietzfelbinger, How Good Is Multi-Pivot Quicksort?, ACM Transactions on Algorithms (TALG), Volume 13 Issue 1, 2016.
M. Aumüller and M. Dietzfelbinger, How Good Is Multi-Pivot Quicksort?, arXiv:1510.04676 [cs.DS], 2016.
Daniel Krenn, Quickstar, Program in SageMath, on GitHub.
Index entries for sequences related to sorting.

Cf. A001768, A003070, A036604, A117627, A117628, A159324, A288964, A288965, A288971.

a(9)-a(11) from Melanie Siebenhofer, Jan 29 2018

a(12)-a(13) from Melanie Siebenhofer, Feb 02 2018

A138772 Number of entries in the second cycles of all permutations of {1,2,...,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Original entry on oeis.org

0, 1, 5, 27, 168, 1200, 9720, 88200, 887040, 9797760, 117936000, 1536796800, 21555072000, 323805081600, 5187108326400, 88268019840000, 1590132031488000, 30233431388160000, 605024315191296000, 12711912992722944000, 279783730940313600000, 6437458713635389440000

Offset: 1

Emeric Deutsch, Apr 10 2008

nonn

			a(3) = 5 because the number of entries in the second cycles of (1)(2)(3), (1)(23), (132), (12)(3), (123) and (13)(2) is 1+2+0+1+0+1=5.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A138771.
Cf. A159324, A000254.
Column k=2 of A185105.

List([1..30], n-> (n-1)*(n+2)*Factorial(n-1)/4); # G. C. Greubel, Jul 07 2019

[(n-1)*(n+2)*Factorial(n-1)/4: n in [1..30]]; // G. C. Greubel, Jul 07 2019

seq((1/4)*factorial(n-1)*(n-1)*(n+2), n = 1 .. 30);

Table[((1/4)(n-1)!(n-1)(n+2)),{n,1,30}] (* Vincenzo Librandi, May 14 2012 *)

vector(30, n, (n-1)*(n+2)*(n-1)!/4) \\ G. C. Greubel, Jul 07 2019

[(n-1)*(n+2)*factorial(n-1)/4 for n in (1..30)] # G. C. Greubel, Jul 07 2019

a(n) = (1/4)*(n-1)!*(n-1)*(n+2).
a(n) = (n+1)*a(n-1) + (n-2)!.
a(n) = (n-1)*a(n-1) + n!/2.
a(n) = Sum_{k=0..n-1} k*A138771(n,k).
E.g.f. if offset 0: x*(2-x)/(2*(1-x)^3). Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010
a(n) = A000254(n-1) + A159324(n-1). - Gary Detlefs, May 13 2012
a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (j/i)). - Pedro Caceres, Apr 19 2019
E.g.f.: ( x*(2-x)/(1-x)^2 + 2*log(1-x) )/4. - G. C. Greubel, Jul 07 2019
D-finite with recurrence a(n) +(-n-1)*a(n-1) -2*a(n-2) +2*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

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A288964 Number of key comparisons to sort all n! permutations of n elements by quicksort.

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A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

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A288970 Number of key comparisons to sort all n! permutations of n elements by the optimal trial-pivot quicksort.

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A138772 Number of entries in the second cycles of all permutations of {1,2,...,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

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A212395 Number of move operations required to sort all permutations of [n] by insertion sort.

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A285231 Number of entries in the third cycles of all permutations of [n].