A008291 -id:A008291 - cp's OEIS frontend

A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.

1, 0, 10, 40, 315, 2464, 22260, 222480, 2447445, 29369120, 381798846, 5345183480, 80177752655, 1282844041920, 21808348713320, 392550276838944, 7458455259940905, 149169105198816960, 3132551209175157490, 68916126601853463240

Offset: 3

N. J. A. Sloane

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
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).

T. D. Noe, Table of n, a(n) for n = 3..100
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
Index entries for sequences related to permutations with fixed points

Cf. A008290.
A diagonal of A008291.
Cf. A170942.

# with k fixed-points:
G:=exp(-z)*z^k/((1-z)*k!: Gser:=series(G,z,21):
for n from k to 20 do a(n)=n!*coeff(Gser,z,n): end do: # Paul Weisenhorn, May 30 2010

Table[Subfactorial[n - 3]*Binomial[n, 3], {n, 3, 22}] (* Zerinvary Lajos, Jul 10 2009 *)

my(x='x+O('x^66)); Vec( serlaplace(exp(-x)/(1-x)*(x^3/3!)) ) \\ Joerg Arndt, Feb 19 2014

A000449_list, m, x = [], 1, 0
for n in range(3,21):
    x, m = x*n + m*(n*(n-1)*(n-2)//6), -m
    A000449_list.append(x) # Chai Wah Wu, Sep 23 2014

a(n) = Sum_{j=2..n-3} (-1)^j*n!/(3!*j!) = A008290(n,3).
For n >= 3 a(n) = C(n, 3) * A000166(n-3) = 1/6 * n! * Sum_{k=0..n-3} (-1)^k/k!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 14 2001
E.g.f.: 1/(exp(x)*(1-x))*(x^3)/6. - Wenjin Woan, Nov 20 2008
E.g.f.: x^3*exp(-x)/(3!*(1-x)). - Geoffrey Critzer, Nov 03 2012
a(n) ~ n! * exp(-1)/6. - Vaclav Kotesovec, Mar 17 2014
a(n) = n*a(n-1) - (-1^n)*n*(n-1)*(n-2)/6, a(n) = 0 for n= 0, 1, 2. - Chai Wah Wu, Sep 23 2014
O.g.f.: (1/6)*Sum_{k>=3} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
D-finite with recurrence (-n+3)*a(n) +n*(n-4)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

A073107 Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 16, 15, 6, 1, 65, 64, 30, 8, 1, 326, 325, 160, 50, 10, 1, 1957, 1956, 975, 320, 75, 12, 1, 13700, 13699, 6846, 2275, 560, 105, 14, 1, 109601, 109600, 54796, 18256, 4550, 896, 140, 16, 1, 986410, 986409, 493200, 164388, 41076, 8190, 1344, 180, 18, 1

Offset: 0

Vladeta Jovovic, Aug 19 2002

Triangle is second binomial transform of A008290. - Paul Barry, May 25 2006

Ignoring signs, n-th row is the coefficient list of the permanental polynomial of the n X n matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 02 2012

			exp((1 + y)*x)/(1 - x) =
  1 +
  1/1! * (2 + y) * x +
  1/2! * (5 + 4*y + y^2) * x^2 +
  1/3! * (16 + 15*y + 6*y^2 + y^3) * x^3 +
  1/4! * (65 + 64*y + 30*y^2 + 8*y^3 + y^4) * x^4 +
  1/5! * (326 + 325*y + 160*y^2 + 50*y^3 + 10*y^4 + y^5) * x^5 + ...
Triangle starts:
  [0]     1;
  [1]     2,     1;
  [2]     5,     4,    1;
  [3]    16,    15,    6,    1;
  [4]    65,    64,   30,    8,   1;
  [5]   326,   325,  160,   50,  10,   1;
  [6]  1957,  1956,  975,  320,  75,  12,  1;
  [7] 13700, 13699, 6846, 2275, 560, 105, 14, 1;

Seiichi Manyama, Rows n = 0..139, flattened
Wikipedia, Sheffer sequence.

Cf. A008290, A008291, A046802, A093375 (unsigned inverse), A094587, A010842 (row sums), A000142 (alternating row sums), A367963 (central terms).

Column k=0..4 give A000522, A007526, A038155, A357479, A357480.

T := (n, k) -> binomial(n,k)*KummerU(k-n, k-n, 1);
seq(seq(simplify(T(n, k)), k = 0..n), n=0..8);  # Peter Luschny, Oct 16 2024

perm[m_List] := With[{v=Array[x,Length[m]]},Coefficient[Times@@(m.v),Times@@v]] ;
A[q_] := Array[KroneckerDelta[#1,#2] + 1&,{q,q}] ;
n = 1 ; Print[{1}]; While[n < 10, Print[Abs[CoefficientList[perm[A[n] - IdentityMatrix[n] * k], k]]]; n++] (* John M. Campbell, Jul 02 2012 *)
A073107[n_, k_] := If[n == k, 1, Floor[E*(n - k)!]*Binomial[n, k]];
Table[A073107[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Oct 16 2024 *)

def T(n, k):
    return sum(binomial(j,k) * factorial(n) // factorial(j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(n+1)])
# Peter Luschny, Oct 16 2024

O.g.f. for k-th column is (1/k!)*Sum_{i >= k} i!*x^i/(1-x)^(i+1).
For n > 0, T(n, 0) = floor(n!*exp(1)) = A000522(n), T(n, 1) = floor(n!*exp(1) - 1) = A007526(n), T(n, 2) = 1/2!*floor(n!*exp(1) - 1 - n) = A038155(n), T(n, 3) = 1/3!*floor(n!*exp(1) - 1 - n - n*(n - 1)), T(n, 4) = 1/4!*floor(n!*exp(1) - 1 - n - n*(n - 1) - n*(n - 1)*(n - 2)), ... .
Row sums give A010842.
E.g.f. for k-th column is (x^k/k!)*exp(x)/(1 - x).
O.g.f. for k-th row is n!*Sum_{k = 0..n} (1 + x)^k/k!.
T(n,k) = Sum_{j = 0..n} binomial(j,k)*n!/j!. - Paul Barry, May 25 2006
-exp(-x) * Sum_{k=0..n} T(n,k)*x^k = Integral (x+1)^n*exp(-x) dx = -exp(1)*Gamma(n+1,x+1). - Gerald McGarvey, Mar 15 2009
From Peter Bala, Sep 20 2012: (Start)
Exponential Riordan array [exp(x)/(1-x),x] belonging to the Appell subgroup, which factorizes in the Appell group as [1/1-x,x]*[exp(x),x] = A094587*A007318.
The n-th row polynomial R(n,x) of the triangle satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x + y) = Sum {k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.
Matrix inverse of triangle is a signed version of A093375. (End)
From Tom Copeland, Oct 20 2015: (Start)
The raising operator, with D = d/dx, for the row polynomials is RP = x + d{log[e^D/(1-D)]}/dD = x + 1 + 1/(1-D) =  x + 2 + D + D^2 + ..., i.e., RP R(n,x) = R(n+1,x).
This operator is the limit as t tends to 1 of the raising operator of the polynomials p(n,x;t) described in A046802, implying R(n,x) = p(n,x;1). Compare with the raising operator of A094587, x + 1/(1-D), and that of signed A093375, x - 1 - 1/(1-D).
From the Appell formalism, the row polynomials RI(n,x) of signed A093375 are the umbral inverse of this entry's row polynomials; that is, R(n,RI(.,x)) = x^n = RI(n,R(.,x)) under umbral composition. (End)
From Werner Schulte, Sep 07 2020: (Start)
T(n,k) = (n! / k!) * (Sum_{i=k..n} 1 / (n-i)!) for 0 <= k <= n.
T(n,k) = n * T(n-1,k) + binomial(n,k) for 0 <= k <= n with initial values T(0,0) = 1 and T(i,j) = 0 if j < 0 or j > i.
T(n,k) = A000522(n-k) * binomial(n,k) for 0 <= k <= n. (End)

More terms from Emeric Deutsch, Feb 23 2004

A164863 Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678514, 4211825, 27602602, 190077045, 1368705291, 10254521370, 79527284317, 635182667816, 5199414528808, 43426867585575, 368654643520692, 3170300933550687, 27542984610086665, 241205285284001240

Offset: 0

Alois P. Heinz, Aug 28 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Moreira, N.; Reis, R. "On the Density of Languages Representing Finite Set Partitions", Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Eric Weisstein's World of Mathematics, Set Partition
Index entries for linear recurrences with constant coefficients, signature (37, -574, 4858, -24409, 74053, -131256, 122652, -45360).

Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164864.

A row of the array in A278984.

# first program:
a:= n-> ceil(103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880): seq(a(n), n=0..25);
# second program:
a:= n-> add(Stirling2(n, k), k=0..9): seq(a(n), n=0..25);

Table[Sum[StirlingS2[n, k], {k, 0, 9}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)

a(n) = Sum_{k=0..9} stirling2 (n,k).
a(n) = ceiling (103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880).
G.f.: (16687*x^8 -67113*x^7 +88620*x^6 -56993*x^5 +20529*x^4 -4353*x^3 +539*x^2 -36*x+1) / ((9*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=9. - Robert A. Russell, Apr 25 2018

A060008 a(n) = 9binomial(n,4) = 3n(n-1)(n-2)(n-3)/8.

Original entry on oeis.org

0, 0, 0, 0, 9, 45, 135, 315, 630, 1134, 1890, 2970, 4455, 6435, 9009, 12285, 16380, 21420, 27540, 34884, 43605, 53865, 65835, 79695, 95634, 113850, 134550, 157950, 184275, 213759, 246645, 283185, 323640, 368280, 417384, 471240, 530145, 594405

Offset: 0

Henry Bottomley, Mar 16 2001

Number of permutations of n letters where exactly four change position.

			a(6) = 135 since there are 15 ways to choose the four points that move and 9 ways to move them and 15*9 = 135.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

For changing 0, 1, 2, 3, 4, 5, n-4, n elements see A000012, A000004, A000217 (offset), A007290, A060008, A060836, A000475, A000166. Also see A000332, A008290.

A diagonal of A008291.

9*Binomial[Range[0,40],4] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,0,9},40] (* Harvey P. Dale, Jun 09 2014 *)

a(n) = { 3*n*(n - 1)*(n - 2)*(n - 3)/8 } \\ Harry J. Smith, Jul 01 2009

Equals 3*A050534. - Zerinvary Lajos, Feb 12 2007
G.f.: 9*x^4/(1-x)^5. - Colin Barker, Jul 02 2012
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 4/27.
Sum_{n>=4} (-1)^n/a(n) = 32*log(2)/9 - 64/27. (End)

A129135 Number of permutations of [n] with exactly 5 fixed points.

Original entry on oeis.org

1, 0, 21, 112, 1134, 11088, 122430, 1468368, 19090071, 267258992, 4008887883, 64142201760, 1090417436108, 19627513841376, 372922762997772, 7458455259939936, 156627560458759005, 3445806330092671776, 79253545592131484497, 1902085094211155585424

Offset: 5

Zerinvary Lajos, May 25 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 5..200
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
Index entries for sequences related to permutations with fixed points

Cf. A008290, A008291, A170942.

a:=n->sum((n-1)!*sum((-1)^k/(k-4)!, j=0..n-1), k=4..n-1)/5!: seq(a(n), n=5..24);
x:='x'; G(x):=exp(-x)/(1-x)*(x^5/5!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=5..24); # Zerinvary Lajos, Apr 03 2009
a:= n-> simplify(pochhammer(6, n-5)*GAMMA(n-4, -1)*exp(-1)/GAMMA(n-4)):
seq(a(n), n = 5 .. 24); # Miles Wilson, Aug 04 2024

With[{nn=30},Drop[CoefficientList[Series[Exp[-x]/(1-x) x^5/5!,{x,0,nn}],x]Range[0,nn]!,5]] (* Harvey P. Dale, Jan 22 2013 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ Joerg Arndt, Feb 17 2014

from sympy import binomial
A129135_list, m, x = [], 1, 0
for n in range(5,21):
    x, m = x*n + m*binomial(n,5), -m
    A129135_list.append(x) # Chai Wah Wu, Nov 01 2014

a(n) = A008290(n,5).
E.g.f.: exp(-x)/(1-x)*(x^5/5!). - Zerinvary Lajos, Apr 03 2009
a(n) = n*a(n-1) - (-1^n)*binomial(n,5) with a(n) = 0 for n = 0,1,2,3,4. - Chai Wah Wu, Nov 01 2014
D-finite with recurrence (-n+5)*a(n) +n*(n-6)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015
O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Offset corrected by Susanne Wienand, Feb 17 2014

A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 4, 0, 5, 6, 10, 2, 1, 21, 36, 42, 12, 9, 0, 117, 226, 219, 104, 47, 6, 1, 792, 1568, 1472, 800, 328, 64, 16, 0, 6205, 12360, 11596, 6652, 2658, 688, 148, 12, 1, 55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25, 0, 543597, 1085560, 1030649, 614420, 255830, 77732, 17750, 2876, 365, 20, 1

Offset: 0

Alois P. Heinz, Jan 23 2019

			T(4,0) = 5: 1234, 1432, 3214, 3412, 4231.
T(4,1) = 6: 2431, 3241, 3421, 4132, 4213, 4312.
T(4,2) = 10: 1243, 1324, 1342, 1423, 2134, 2314, 2413, 3124, 3142, 4321.
T(4,3) = 2: 2341, 4123.
T(4,4) = 1: 2143.
Triangle T(n,k) begins:
      1;
      1,      0;
      1,      0,      1;
      2,      0,      4,     0;
      5,      6,     10,     2,     1;
     21,     36,     42,    12,     9,    0;
    117,    226,    219,   104,    47,    6,    1;
    792,   1568,   1472,   800,   328,   64,   16,   0;
   6205,  12360,  11596,  6652,  2658,  688,  148,  12,  1;
  55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25,  0;
  ...

Alois P. Heinz, Rows n = 0..24, flattened

Column k=0 gives A078480.
Row sums give A000142.
Main diagonal gives A059841.
Cf. A008290, A008291, A052582, A082033, A323671.

b:= proc(s) option remember; expand((n-> `if`(n=0, 1, add(
     `if`(abs(n-j)=1, x, 1)*b(s minus {j}), j=s)))(nops(s)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n})):
seq(T(n), n=0..12);

b[s_] := b[s] = Expand[With[{n = Length[s]}, If[n==0, 1, Sum[
     If[Abs[n-j]==1, x, 1]*b[s~Complement~{j}], {j, s}]]]];
T[n_] := PadRight[CoefficientList[b[Range[n]], x], n+1];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A052582(n-1) for n > 0.

Sum_{k=0..n} (k+1) * T(n,k) = A082033(n-1) for n > 0.

A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1

Offset: 0

Alois P. Heinz, Dec 19 2021

			T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: |A069856|, A348590.
Row sums give A000312.
T(n+1,n-1) gives A045943.
Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A055897(n).
Sum_{k=1..n} T(n,k) = A350134(n).
From Mélika Tebni, Nov 24 2022: (Start)
T(n,k) = binomial(n, k)*|A069856(n-k)|.
E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

A129218 Permutations with exactly 10 fixed points.

Original entry on oeis.org

1, 0, 66, 572, 9009, 132132, 2122120, 36056592, 649062414, 12332093488, 246642054516, 5179482792120, 113948622073286, 2620818306541512, 62899639358957544, 1572490983970669840, 40884765583242727575

Offset: 10

Zerinvary Lajos, May 25 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 10..200

Cf. A008290, A008291, A170942.

a:=n->sum(n!*sum((-1)^k/(k-9)!, j=0..n), k=9..n): seq(-a(n)/10!, n=9..27);
restart: G(x):=exp(-x)/(1-x)*(x^10/10!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=10..26); # Zerinvary Lajos, Apr 03 2009

With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^10/10!, {x, 0, nn}], x]Range[0, nn]!, 10]] (* Vincenzo Librandi, Feb 19 2014 *)

x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^9/9!)) ) \\ Joerg Arndt, Feb 19 2014

a(n) = A008290(n,10).
E.g.f.: exp(-x)/(1-x)*(x^10/10!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/10!)*Sum_{k>=10} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Changed offset from 0 to 10 by Vincenzo Librandi, Feb 19 2014

Edited by Joerg Arndt, Feb 19 2014

A060836 Number of permutations of n letters where exactly 5 change position.

Original entry on oeis.org

0, 0, 0, 0, 44, 264, 924, 2464, 5544, 11088, 20328, 34848, 56628, 88088, 132132, 192192, 272272, 376992, 511632, 682176, 895356, 1158696, 1480556, 1870176, 2337720, 2894320, 3552120, 4324320, 5225220, 6270264, 7476084, 8860544, 10442784, 12243264, 14283808, 16587648

Offset: 1

Robert Goodhand (rgoodhand(AT)hotmail.com), May 12 2001

			a(8) = a(7) * 8/(8-5) = 924 * 8/3 = 2464.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

For changing 0, 1, 2, 3, 4, 5, n-4, n elements see A000012, A000004, A000217 (offset), A007290, A060008, A060836, A000475, A000166. Also see A000332, A008290.
Rencontre sequences are A000166 A000240 A000387 A000449 and A000475.
A diagonal of A008291.

a(n) = { 44*binomial(n, 5) } \\ Harry J. Smith, Jul 19 2009

a(n) = 44*binomial(n, 5).
a(n) = a(n-1)*n/(n-5).
G.f.: 44*x^5/(1 - x)^6. - Colin Barker, Apr 22 2012

A129136 Permutations with exactly 6 fixed points.

Original entry on oeis.org

1, 0, 28, 168, 1890, 20328, 244860, 3181464, 44543499, 668147480, 10690367688, 181736238320, 3271252308324, 62153793831024, 1243075876659240, 26104593409789776, 574301055015449685, 13208924265355241808

Offset: 6

Zerinvary Lajos, May 25 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 6..200
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
Index entries for sequences related to permutations with fixed points

Cf. A008290, A008291, A170942.

a:=n->sum(n!*sum((-1)^k/(k-5)!, j=0..n), k=5..n): seq(-a(n)/6!, n=5..24);
restart: G(x):=exp(-x)/(1-x)*(x^6/6!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=6..23); # Zerinvary Lajos, Apr 03 2009

With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^6/6!, {x, 0, nn}], x]Range[0, nn]!, 6]] (* Vincenzo Librandi, Feb 19 2014 *)

x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^6/6!)) ) \\ Joerg Arndt, Feb 19 2014

a(n) = A008290(n,6).
E.g.f.: exp(-x)/(1-x)*(x^6/6!). [Zerinvary Lajos, Apr 03 2009]
O.g.f.: (1/6!)*Sum_{k>=6} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017
D-finite with recurrence +(-n+6)*a(n) +n*(n-7)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Changed offset from 0 to 6 by Vincenzo Librandi, Feb 19 2014

Edited by Joerg Arndt, Feb 19 2014

cp's OEIS Frontend

A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.

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A073107 Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x).

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A164863 Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.

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A060008 a(n) = 9*binomial(n,4) = 3n*(n-1)*(n-2)*(n-3)/8.

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A129135 Number of permutations of [n] with exactly 5 fixed points.

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A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A060008 a(n) = 9binomial(n,4) = 3n(n-1)(n-2)(n-3)/8.