A052881 -id:A052881 - cp's OEIS frontend

A000276 Associated Stirling numbers.

3, 20, 130, 924, 7308, 64224, 623376, 6636960, 76998240, 967524480, 13096736640, 190060335360, 2944310342400, 48503818137600, 846795372595200, 15618926924697600, 303517672703078400, 6198400928176128000, 132720966600284160000, 2973385109386137600000

Offset: 4

N. J. A. Sloane

nonn

a(n) is also the number of permutations of n elements, without any fixed point, with exactly two cycles. - Shanzhen Gao, Sep 15 2010

			a(4) = 3 because we have: (12)(34),(13)(24),(14)(23). - _Geoffrey Critzer_, Nov 03 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
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).
Shanzhen Gao, Permutations with Restricted Structure (in preparation).

Alois P. Heinz, Table of n, a(n) for n = 4..150

A diagonal of triangle in A008306.

Cf. A052518, A052881, A259456.

nn=25;a=Log[1/(1-x)]-x;Drop[Range[0,nn]!CoefficientList[Series[a^2/2,{x,0,nn}],x],4]  (* Geoffrey Critzer, Nov 03 2012 *)
a[n_] := (n-1)!*(HarmonicNumber[n-2]-1); Table[a[n], {n, 4, 23}] (* Jean-François Alcover, Feb 06 2016, after Gary Detlefs *)

a(n) = (n-1)!*sum(i=2, n-2, 1/i); \\ Michel Marcus, Feb 06 2016

a(n) = (n-1)!*Sum_{i=2..n-2} 1/i = (n-1)!*(Psi(n-1)+gamma-1). - Vladeta Jovovic, Aug 19 2003
With alternating signs: Ramanujan polynomials psi_3(n-2, x) evaluated at 1. - Ralf Stephan, Apr 16 2004
E.g.f.: ((x+log(1-x))^2)/2. [Corrected by Vladeta Jovovic, May 03 2008]
a(n) = Sum_{i=2..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 15 2010
a(n) = (n+3)!*(h(n+2)-1), with offset 0, where h(n)=sum(1/k,k=1..n). - Gary Detlefs, Sep 11 2010
Conjecture: (-n+2)*a(n) +(n-1)*(2*n-5)*a(n-1) -(n-1)*(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 18 2015
Conjecture: a(n) +2*(-n+2)*a(n-1) +(n^2-6*n+10)*a(n-2) +(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 18 2015
a(n) = A000254(n-1) - (n-1)! - (n-2)!. - Anton Zakharov, Sep 24 2016

More terms from Christian G. Bower

A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

Original entry on oeis.org

2, 3, 6, 8, 12, 24, 30, 40, 60, 120, 144, 180, 240, 360, 720, 840, 1008, 1260, 1680, 2520, 5040, 5760, 6720, 8064, 10080, 13440, 20160, 40320, 45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880, 403200, 453600, 518400, 604800, 725760, 907200, 1209600, 1814400, 3628800

Offset: 2

Leroy Quet, Dec 07 2000

Together with 1, numbers n such that n divides k! if and only if k! >= n. - Charles R Greathouse IV, Aug 16 2016

			Triangle begins:
      2;
      3,     6;
      8,    12,    24;
     30,    40,    60,   120;
    144,   180,   240,   360,   720;
    840,  1008,  1260,  1680,  2520,   5040;
   5760,  6720,  8064, 10080, 13440,  20160,  40320;
  45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Florian Unger and Jonathan Krebs, MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs, arXiv:2309.02938 [math.ST], 2023.

Columns k=1..5 are A001048(n-1), A052747, A052759, A052778, A052794.
Row sums are A052881.
Cf. A019739. A077012.

Flatten[Table[n!/(n-k),{n,2,10},{k,n-1}]] (* Harvey P. Dale, Jul 23 2014 *)

PARI
```
T(n,k)={if(kAndrew Howroyd, Aug 08 2020
```

Sum_{n>=2} Sum_{k=1..n-1} 1/T(n, k) = e/2 (A019739). - Amiram Eldar, Jun 29 2025

A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

Original entry on oeis.org

-1, 2, -2, -6, 9, -3, 24, -44, 24, -4, -120, 250, -175, 50, -5, 720, -1644, 1350, -510, 90, -6, -5040, 12348, -11368, 5145, -1225, 147, -7, 40320, -104544, 105056, -54152, 15680, -2576, 224, -8, -362880, 986256, -1063116, 605556, -202041, 40824, -4914, 324, -9, 3628800, -10265760, 11727000, -7236800

Offset: 1

Vladeta Jovovic, Jun 05 2004

			Triangle starts:
[n\k    1        2       3      4      5      6     7  8]
[1]    -1;
[2]     2,      -2;
[3]    -6,       9,     -3;
[4]    24,     -44,     24,     -4;
[5]  -120,     250,   -175,     50,    -5;
[6]   720,   -1644,   1350,   -510,    90,    -6;
[7] -5040,   12348, -11368,   5145, -1225,   147,   -7;
[8] 40320, -104544, 105056, -54152, 15680, -2576,  224,  -8;

Cf. A019538, A028246, A163626.

Cf. A000142, A052881.

T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1);
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, May 26 2020

Table[StirlingS1[n+1,k]-StirlingS1[n,k-1],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 25 2024 *)

E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]
The matrix inverse of the Worpitzky triangle. More precisely:
T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - Peter Luschny, May 26 2020

Offset of k shifted and edited by Peter Luschny, May 26 2020

A087301 a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.

Original entry on oeis.org

2, 3, 20, 70, 564, 3108, 30624, 230256, 2705760, 25771680, 352805760, 4067556480, 63651813120, 861371884800, 15176802816000, 235775183616000, 4620563523072000, 81032645804544000, 1748700390205440000

Offset: 2

Vladeta Jovovic, Oct 20 2003

nonn

Stirling transform of A052882(n)=[0,2,9,52,375,...] is a(n+1)=[0,2,3,20,...]. - Michael Somos, Mar 04 2004

Cf. A024167, A052881.

Rest[Table[n!Sum[(-1)^(i+1)/i,{i,n-1}],{n,20}]] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=if(n<0,0,n!*polcoeff(log(1+x+x*O(x^n))*x/(1-x),n))

E.g.f.: x*log(1+x)/(1-x). a(n) = 1/2*(-1)^n*n!*(2*(-1)^n*log(2)+Psi(1/2+1/2*n)-Psi(1/2*n)).

a(n) ~ n! * log(2). - Vaclav Kotesovec, Jul 01 2018

A278463 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 3, 9, 4, 4, 36, 44, 12, 5, 110, 355, 250, 48, 6, 300, 2010, 3480, 1644, 240, 7, 777, 9625, 32235, 35728, 12348, 1440, 8, 1960, 42056, 242200, 498512, 390880, 104544, 10080, 9, 4860, 173754, 1605744, 5466321, 7745220, 4581036, 986256, 80640, 10, 11880, 691620, 9807840, 51506490, 117711720, 123330680, 57537360, 10265760, 725760

Offset: 1

Gheorghe Coserea, Jan 18 2017

			A(x;t) = x + (2*t+2)*x^2/2! + (3*t^2+9*t+4)*x^3/3! + (4*t^3+36*t^2+44*t+12)*x^4/4! + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
[1]  1;
[2]  2,       2;
[3]  3,       9,       4;
[4]  4,       36,      44,      12;
[5]  5,       110,     355,     250,     48;
[6]  6,       300,     2010,    3480,    1644,    240;
[7]  7,       777,     9625,    32235,   35728,   12348,   1440;
[8]  ...

Gheorghe Coserea, Rows n = 1..101, flattened.
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

N=11; x = 'x+O('x^N);
concat(apply(p->Vec(p), Vec(serlaplace((t-1)*log(1-x) - log(-x + exp(t*log(1-x))) - x))))

A(x;t) = Sum {n>=1} P_n(t)*x^n/n! = (t-1)*log(1-x) - log(-x + exp(t*log(1-x))) - x.
A278458(x;t) = serreverse(A(-x;t))(-x).
A098558(n-1) = P_n(0), A032184(n) = P_n(1).
A052881(n) = T(n,n-1).

cp's OEIS Frontend

A000276 Associated Stirling numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A087301 a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A278463 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Views

Author

Keywords

Examples

Links

Programs

PARI

Formula