A001048 -id:A001048 - cp's OEIS frontend

A108217 a(0) = 1, a(1) = 1, a(n) = n! + (n-2)! for n >= 2.

1, 1, 3, 7, 26, 126, 744, 5160, 41040, 367920, 3669120, 40279680, 482630400, 6266937600, 87657292800, 1313901388800, 21009968179200, 356995102464000, 6423296495616000, 122000787836928000, 2439304381882368000, 51212587272118272000, 1126433629785784320000

Offset: 0

Miklos Kristof, following a suggestion from Peter Boros, (borospet(AT)freemail.hu), Jun 16 2005

In factorial base representation (A007623) the terms of this sequence look as: 1, 1, 11, 101, 1010, 10100, 101000, ... From a(3)=7 onward each term begins always with "101", which is then followed by n-3 zeros. - Antti Karttunen, Sep 23 2016

			a(6) = 6!+4! = 720+24 = 744.

Index entries for sequences related to factorial base representation

Cf. A000142, A002061, A007623, A001048, A030495.

Row 5 of A276955, from term a(3)=7 onward.

a:= n-> `if`(n<2, 1, n!+(n-2)!):
seq(a(n), n=0..30);

Table[If[n<2,1,n!+(n-2)!],{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
Join[{1,1},#[[1]]+#[[3]]&/@Partition[Range[0,20]!,3,1]] (* Harvey P. Dale, Nov 19 2015 *)

(define (A108217 n) (if (<= n 1) 1 (* (A002061 n) (A000142 (- n 2))))) ;; Antti Karttunen, Sep 23 2016

For n >= 2, a(n) = A002061(n) * (n-2)! - Antti Karttunen, Sep 23 2016

E.g.f.: x + (1-x)*log(1-x) + 1/(1-x). - Andrew Howroyd, May 09 2021

Corrected by Georg Fischer, May 09 2021

A226198 a(n) = floor((n-1)!/n).

Original entry on oeis.org

1, 0, 0, 1, 4, 20, 102, 630, 4480, 36288, 329890, 3326400, 36846276, 444787200, 5811886080, 81729648000, 1230752346352, 19760412672000, 336967037143578, 6082255020441600, 115852476579840000, 2322315553259520000, 48869596859895986086

Offset: 1

Vincenzo Librandi, May 31 2013

nonn

(h-1)!/h is integer if h belongs to A056653.

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

Cf. A001048 ((n+1)!/n), A056653, A091330 (floor((p-1)!/p), where p is prime), A175787 (prime numbers together with 4).

[Floor(Factorial(n-1)/n): n in [1..25]];

Mathematica
```
Table[Quotient[(n - 1)!, n], {n, 25}]
```

Edited by Bruno Berselli, May 31 2013

A304036 Number of partitions of n into at most 2 copies of 1!, 3 copies of 2!, 4 copies of 3!, ... .

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2

Offset: 0

Seiichi Manyama, May 05 2018

			a(6) = 3 because we have [6], [2,2,2] and [2,2,1,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Factorial number system

Cf. A001048, A064986, A303939, A304039.

G.f.: Product_{j>=1} Sum_{k=0..j+1} x^(k*j!) = Product_{j>=1} (1-x^((j+1)!+j!))/(1-x^(j!)).

A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

Original entry on oeis.org

1, 1, 2, 3, 3, 10, 6, 8, 25, 45, 20, 30, 176, 60, 250, 90, 144, 721, 861, 770, 1344, 504, 840, 6406, 1778, 7980, 6300, 8736, 3360, 5760, 42561, 23283, 38808, 75348, 45360, 66240, 25920, 45360, 436402, 84150, 363680, 456120, 708048, 378000, 572400, 226800, 403200

Offset: 0

Alois P. Heinz, Aug 14 2023

T(0,0) = 1 by convention.

			T(4,0) = 10: (1)(2)(3)(4), (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), (1432).
T(4,1) = 6: (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4), (14)(2)(3).
T(4,2) = 8: (1)(234), (1)(243), (123)(4), (132)(4), (124)(3), (142)(3), (134)(2), (143)(2).
Triangle T(n,k) begins:
      1;
      1;
      2;
      3,     3;
     10,     6,     8;
     25,    45,    20,    30;
    176,    60,   250,    90,   144;
    721,   861,   770,  1344,   504,   840;
   6406,  1778,  7980,  6300,  8736,  3360,  5760;
  42561, 23283, 38808, 75348, 45360, 66240, 25920, 45360;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Wikipedia, Permutation

Row sums give A000142.
Column k=0 gives A005225 (for n>=1).
T(n+1,n-1) gives A001048(n) (for n>=1).
Cf. A126074, A145877, A364971, A365229.

b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
     b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..12);

b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

T(n,k) == 0 (mod k!).

Sum_{k=0..max(0,n-2)} T(n,k)/k! = A365229(n).

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

A092824 Farey-factorial numerators.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 30, 40, 48, 60, 72, 80, 90, 96, 120, 144, 180, 240, 288, 360, 432, 480, 540, 576, 600, 720, 840, 1008, 1260, 1440, 1680, 2016, 2160, 2520, 2880, 3024, 3360, 3600, 3780, 4032, 4200, 4320, 5040, 5760, 6720, 8064, 10080

Offset: 1

Clark Kimberling, Mar 06 2004

nonn

The last number in the n-th segment is n!. Let f(n) be the first number in segment n; except for initial terms, f is A001048 and A059171. Let g(n) be the second number in segment n; except for initial terms, g is A052747. Except for the initial terms, the number of numbers in segment n is given by A015614.

			The sequence begins with these segments:
  1
  2
  3 4 6
  8 12 16 18 24
For the next segment, start with these Farey fractions of order 5:
  1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/5.
Multiply these by 5! to get
  30 40 48 60 72 80 90 96 120.

Cf. A000142, A001048, A015614, A059171, A052747.

f[n_] := n! * Table[a/b, {b, 1, n}, {a, 1, b}] // Flatten // Union // Rest; Flatten[Table[f[n], {n, 1, 8}] /. {} -> {1}][[1 ;; 51]] (* Jean-François Alcover, May 18 2011 *)

Let S(n) be the set of integers an!/b, where a/b ranges through the positive Farey fractions of order n. A092824 is the increasing sequence of integers in the union of the sets S(n), for n>=1.

A174549 a(n) = (2n-1)! + (2n)!.

Original entry on oeis.org

3, 30, 840, 45360, 3991680, 518918400, 93405312000, 22230464256000, 6758061133824000, 2554547108585472000, 1175091669949317120000, 646300418472124416000000, 418802671169936621568000000, 315777214062132212662272000000, 274094621805930760590852096000000

Offset: 1

Paul Curtz, Mar 22 2010

x*cos(x) - sin(x) = Sum_{n>=1} (-1)^n/a(n) * x^(2*n+1). - James R. Buddenhagen, Nov 21 2013

Also the number of adjacency matrices for the n-helm graph. - Eric W. Weisstein, May 25 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Adjacency Matrix.
Eric Weisstein's World of Mathematics, Helm Graph.

Cf. A001048, A000255, A068985, A143624, A165457.

[Factorial(2*n-1) + Factorial(2*n): n in [1..15]]; // Vincenzo Librandi, Aug 04 2011

A174549 := proc(n) (1+2*n)*(2*n-1)! ; end proc: # R. J. Mathar, Jan 13 2011

Table[(2*n - 1)! + (2*n)!, {n, 15}] (* T. D. Noe, Nov 21 2013 *)

a(n)=(2*n-1)!*(2*n+1) \\ Charles R Greathouse IV, Nov 21 2013

a(n) = A001048(2n) = (1+2n)*(2n-1)! = 3*A165457(n-1).
Sum_{n>=1} 1/a(n) = A068985 = 1/e = lim_{n->infinity} A000255(n-1)/A001048(n).
zeta(2*n+1) = Integral_{u=0..Pi/2} (sin(u)*log(sin(u))^(2*n+1)/(cos(u)^3))*(-2)^(2*n+1)/(n*a(n)) du. Verified for n=1 to 4 on Wolfram Alpha. - Jean-Claude Babois, Oct 28 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = sin(1)-cos(1) = (-1)*A143624. - Amiram Eldar, Apr 12 2021

A162971 Triangle read by rows: T(n,k) is number of non-derangement permutations of {1,2,...,n} having k cycles (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 8, 6, 1, 0, 30, 35, 10, 1, 0, 144, 210, 85, 15, 1, 0, 840, 1414, 735, 175, 21, 1, 0, 5760, 10752, 6664, 1960, 322, 28, 1, 0, 45360, 91692, 64764, 22449, 4536, 546, 36, 1, 0, 403200, 869040, 679580, 268380, 63273, 9450, 870, 45, 1, 0, 3991680, 9074736, 7704180, 3382280, 902055, 157773, 18150, 1320, 55, 1

Offset: 1

Emeric Deutsch, Jul 22 2009

Sum of entries in row n = A002467(n) (the number of non-derangement permutations of {1,2,...,n}).
T(n,2) = n*(n-2)! = A001048(n-1) for n>=3.
Sum_{k=1..n} k*T(n,k) = A162972(n).

			T(4,2) = 8 because we have (1)(234), (1)(243), (134)(2), (143)(2), (124)(3), (142)(3), (123)(4), and (132)(4).
Triangle starts:
  1;
  0,   1;
  0,   3,   1;
  0,   8,   6,   1;
  0,  30,  35,  10,   1;
  0, 144, 210,  85,  15,   1;
  ...

Alois P. Heinz, Rows n = 1..150, flattened

Cf. A001048, A002467, A162972.

G := (1-exp(-t*z))/(1-z)^t: Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, t, add(expand((j-1)!*
      b(n-j, `if`(j=1, 1, t))*x)*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 15 2023

b[n_, t_] := b[n, t] = If[n == 0, t, Sum[Expand[(j - 1)!*b[n - j, If[j == 1, 1, t]]*x]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := CoefficientList[b[n, 0]/x, x];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 04 2024, after Alois P. Heinz *)

E.g.f.: G(t,z) = (1-exp(-tz))/(1-z)^t.

A162991 The first right hand column of triangle A162990.

Original entry on oeis.org

4, 9, 64, 900, 20736, 705600, 33177600, 2057529600, 162570240000, 15933509222400, 1896219279360000, 269276305858560000, 44970736429301760000, 8724552309817344000000, 1945613940877311344640000

Offset: 1

Johannes W. Meijer, Jul 21 2009

A001048(n) equals the square root of a(n).

A162992 is the second right hand column of triangle A162990.

Array[((#+1)!/#)^2 &, 20] (* Paolo Xausa, Mar 31 2024 *)

a(n) = ((n+1)!/n)^2 for n = 1, 2, 3, ... .

A171005 a(n) = (n+1)*(n-1)!/2.

Original entry on oeis.org

4, 15, 72, 420, 2880, 22680, 201600, 1995840, 21772800, 259459200, 3353011200, 46702656000, 697426329600, 11115232128000, 188305108992000, 3379030566912000, 64023737057280000, 1277273554292736000, 26761922089943040000, 587545834974658560000, 13488008733331292160000

Offset: 3

N. J. A. Sloane, Sep 02 2010

A wheel graph is a graph with n+1 vertices (n>=3) formed by connecting a single vertex to all vertices of an n-cycle. a(n) is the number of labeled wheel graphs. - Geoffrey Critzer, Feb 02 2014

			For n >= 1, the sequence is 1, 3/2, 4, 15, 72, 420, 2880, 22680, 201600, 1995840, ...

Equals A001048/2.

Table[((n+1)*(n-1)!)/2,{n,3,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
nn=20;Drop[Range[0,nn]!CoefficientList[Series[x (Log[1/(1-x)]/2+x^2/4+x/2),{x,0,nn}],x],4] (* Geoffrey Critzer, Feb 02 2014 *)

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(j+1)^(n+1)/(n+1). - Vladimir Kruchinin, Jun 01 2013

D-finite with recurrence -n*a(n) +(n-1)*(n+1)*a(n-1) = 0. - R. J. Mathar, Feb 01 2022

cp's OEIS Frontend

A108217 a(0) = 1, a(1) = 1, a(n) = n! + (n-2)! for n >= 2.

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A226198 a(n) = floor((n-1)!/n).

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A304036 Number of partitions of n into at most 2 copies of 1!, 3 copies of 2!, 4 copies of 3!, ... .

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A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

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Maple

Mathematica

Formula

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

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Mathematica

PARI

Formula

A092824 Farey-factorial numerators.

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Programs

Mathematica

Formula

A174549 a(n) = (2*n-1)! + (2*n)!.

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Magma

Maple

Mathematica

PARI

A174549 a(n) = (2n-1)! + (2n)!.