A000166 -id:A000166 - cp's OEIS frontend

A082491 a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).

1, 0, 2, 12, 216, 5280, 190800, 9344160, 598066560, 48443028480, 4844306476800, 586161043776000, 84407190782745600, 14264815236056985600, 2795903786354347468800, 629078351928420506112000, 161044058093696572354560000, 46541732789077953723039744000

Offset: 0

Emanuele Munarini, Apr 28 2003

a(n) is also the number of pairs of n-permutations p and q such that p(x)<>q(x) for each x in { 1, 2, ..., n }.
Or number of n X n matrices with exactly one 1 and one 2 in each row and column, other entries 0 (cf. A001499). - Vladimir Shevelev, Mar 22 2010
a(n) is approximately equal to (n!)^2/e. - J. M. Bergot, Jun 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).

Cf. A000142, A000166.

with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n!, n=0..15); # Zerinvary Lajos, Jun 11 2008

Table[Subfactorial[n]*n!, {n, 0, 15}] (* Zerinvary Lajos, Jul 10 2009 *)

A000166[0]:1$
A000166[n]:=n*A000166[n-1]+(-1)^n$
  makelist(n!*A000166[n], n, 0, 12); /* Emanuele Munarini, Mar 01 2011 */

d(n)=if(n<1, n==0, n*d(n-1)+(-1)^n);
a(n)=d(n)*n!;
vector(33,n,a(n-1))
/* Joerg Arndt, May 28 2012 */

{a(n) = if( n<2, n==0, n! * round(n! / exp(1)))}; /* Michael Somos, Jun 24 2018 */

A082491_list, m, x = [], 1, 1
for n in range(10*2):
    x, m = x*n**2 + m, -(n+1)*m
    A082491_list.append(x) # Chai Wah Wu, Nov 03 2014

val A082491_pairs: LazyList[BigInt && BigInt] =
  (BigInt(0), BigInt(1)) #::
  (BigInt(1), BigInt(0)) #::
  lift2 {
    case ((n, z), (_, y)) =>
      (n+2, (n+2)*(n+1)*((n+1)*z+y))
  } (A082491_pairs, A082491_pairs.tail)
val A082491: LazyList[BigInt] =
  lift1(_._2)(A082491_pairs)
/** Luc Duponcheel, Jan 25 2020 */

a(n) = n! * d(n) where d(n) = A000166(n).
a(n) = Sum_{k=0..n} binomial(n, k)^2 * (-1)^k * (n - k)!^2 * k!.
a(n+2) = (n+2)*(n+1) * ( a(n+1) + (n+1)*a(n) ).
a(n) ~ 2*Pi*n^(2*n+1)*exp(-2*n-1). - Ilya Gutkovskiy, Dec 04 2016

A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

Original entry on oeis.org

1, 1, -2, 1, -4, 4, 1, -7, 15, -10, 1, -11, 42, -65, 34, 1, -16, 96, -267, 339, -154, 1, -22, 191, -831, 1891, -2103, 874, 1, -29, 344, -2151, 7600, -15023, 15171, -5914, 1, -37, 575, -4880, 24600, -74884, 133147, -124755, 46234, 1, -46, 907, -10025, 68153, -293925, 798564, -1305847, 1151331, -409114, 1, -56

Offset: 1

André F. Labossière, Dec 06 2004

			Subf(9) = [ 9^8 -37*9^7 +575*9^6 -4880*9^5 +24600*9^4 -74884*9^3 +133147*9^2 - 124755*9 +46234 ] = 14833.

Cf. A101560, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A334715 A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 5, 8, 9, 1, 4, 7, 14, 33, 44, 1, 5, 9, 20, 57, 164, 265, 1, 6, 11, 26, 81, 284, 985, 1854, 1, 7, 13, 32, 105, 404, 1705, 6894, 14833, 1, 8, 15, 38, 129, 524, 2425, 11934, 55153, 133496, 1, 9, 17, 44, 153, 644, 3145, 16974, 95473, 496376, 1334961

Offset: 0

Alois P. Heinz, May 08 2020

			Square array A(n,k) begins:
     1,    1,     1,     1,     1,     1,     1,     1, ...
     0,    1,     2,     3,     4,     5,     6,     7, ...
     1,    3,     5,     7,     9,    11,    13,    15, ...
     2,    8,    14,    20,    26,    32,    38,    44, ...
     9,   33,    57,    81,   105,   129,   153,   177, ...
    44,  164,   284,   404,   524,   644,   764,   884, ...
   265,  985,  1705,  2425,  3145,  3865,  4585,  5305, ...
  1854, 6894, 11934, 16974, 22014, 27054, 32094, 37134, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Derangement
Wikipedia, Iverson bracket

Columns k=0-3 give: A000166, A001120, A110043, A110149.
Rows n=0-3 give: A000012, A001477, A005408, A016933.
Main diagonal gives A334716.
Cf. A000142.

A:= proc(n, k) option remember; `if`(n<2,
      (k-1)*n+1, n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := Subfactorial[n] + Boole[n>0] k n!;
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

E.g.f. of column k: (k*exp(x)*x+1)*exp(-x)/(1-x).
A(n,k) = A000166(n) + [n > 0] * (k * n!).
A(n,k) = (k-1)*n + 1 if n<2, A(n,k) = n*A(n-1, k) + (-1)^n if n>=2.

A378157 The least prime dividing !n = A000166(n).

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 11, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 11, 2, 3, 2, 11, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 11, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2

Offset: 3

Amiram Eldar, Nov 18 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A000166, A020639, A152024, A195207, A195208, A195209, A195210, A378158, A378159.

lpf[n_] := Module[{p = 2}, While[! Divisible[n, p], p = NextPrime[p]]; p]; Array[lpf[Subfactorial[#]] &, 50, 3]

lpf(n) = {my(p = 2); while(n % p, p = nextprime(p+1)); p;}
lista(nmax) = {my(s = 1); for(n = 3, nmax, s = n * s + (-1)^n; print1(lpf(s), ", ")); }

a(n) = A020639(A000166(n)).

a(n) = min(A020639(n-1), A378159(n-2)) for n >= 2.

A065087 a(n) = A000166(n)*binomial(n+1,2).

Original entry on oeis.org

0, 0, 3, 12, 90, 660, 5565, 51912, 533988, 6007320, 73422855, 969181620, 13744757598, 208462156812, 3367465610145, 57727981888080, 1046800738237320, 20020064118788592, 402756584036805963, 8502638996332570140, 187953072550509445410, 4341715975916768188740

Offset: 0

N. J. A. Sloane, Nov 10 2001

nonn

a(n) is also the number of permutations of [2n-1] having n-1 isolated fixed points (i.e. adjacent entries are not fixed points). Example: a(2)=3 because we have 132, 213, and 321. - Emeric Deutsch, Apr 18 2009

Seiichi Manyama, Table of n, a(n) for n = 0..400

Equals 3 * A000313(n+2).

Cf. A000166, A000240, A000387, A305730.

a[n_] := Subfactorial[n]*Binomial[n + 1, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 18 2024 *)

a(n) = (n/2)*A000240(n+1). - Zerinvary Lajos, Dec 18 2007, corrected Jul 09 2012
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/2) for n > 1 - Seiichi Manyama, Jun 24 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(2*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018

A173184 Partial sums of A000166.

Original entry on oeis.org

1, 1, 2, 4, 13, 57, 322, 2176, 17009, 150505, 1485466, 16170036, 192384877, 2483177809, 34554278858, 515620794592, 8212685046337, 139062777326001, 2494364438359954, 47245095998005060, 942259727190907181, 19737566982241851721, 433234326593362631602

Offset: 0

Jonathan Vos Post, Feb 12 2010

nonn

Partial sums of subfactorial or rencontres numbers, or derangements (number of permutations of n elements with no fixed points). The subsequence of primes begins: 2, 13, 192384877.

			a(3) = 1 + 0 + 1 + 2 = 4.

Cf. A000166, A001277.

a[0] = 1; a[n_] := a[n] = n*a[n - 1] + (-1)^n; Accumulate@ Array[a, 21, 0] (* Robert G. Wilson v, Apr 01 2011 *)
dr[{n_,a1_,a2_}]:={n+1,a2,n(a1+a2)}; Accumulate[Transpose[NestList[dr,{0,0,1},30]][[3]]] (* Harvey P. Dale, Jul 17 2014 *)
Table[Sum[Subfactorial[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Aug 24 2017 *)

s[0]:1$
s[n]:=n*s[n-1]+(-1)^n$
makelist(sum(s[k],k,0,n),n,0,12); /* Emanuele Munarini, Aug 24 2017 */

G.f.: 1/U(0)/(1-x) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 15 2012
G.f.: 1/(1 - x^2) + (1/(1 - x))*Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
From Emanuele Munarini, Oct 06 2017: (Start)
E.g.f.: exp(-t)/(1-t) - exp(t-2)*(coshIntegral(2-2*t) + sinhIntegral(2-2*t) - expIntegralEi(2)).
a(n+2) - (n+3)*a(n+1) + (n+2)*a(n) = (-1)^n. (End)
D-finite with recurrence a(n+3) - (n+3)*a(n+2) + (n+2)*a(n) = 0. - Emanuele Munarini, Aug 24 2017

A159610 Triangle read by rows, n-th row = n terms of A000255: (1, 3, 11, 53, 309, ...); right border = A000166 starting (1, 2, 9, 44, 265, ...).

Original entry on oeis.org

1, 3, 2, 11, 11, 9, 53, 53, 53, 44, 309, 309, 309, 309, 265, 2119, 2119, 2119, 2119, 2119, 1854, 16687, 16687, 16687, 16687, 16687, 14833, 148329, 148329, 148329, 148329, 148329, 148329, 148329, 133496

Offset: 0

Gary W. Adamson, Apr 17 2009

Row sums = A002469(n+2), representing the game of mousetrap with n cards; where nonzero terms of A002469 start: (1, 5, 31, 203, 1501, ...). A002469(n) = (n-2)*A000255(n-1) + A000166(n). Example 31 = 2*11 + 9 = A002469(4) = 2*A000255(3) + A000166(4).

			First few rows of the triangle:
      1;
      3,     2;
     11,    11,     9;
     53,    53,    53,    44;
    309,   309,   309,   309,   265;
   2119,  2119,  2119,  2119,  2119,  1854;
  16687, 16687, 16687, 16687, 16687, 16687, 14833;
  ...

Cf. A002469, A000166, A000255.

Triangle read by rows, n-th row = n terms of A000255: (1, 3, 11, 53, 309, ...); right border = A000166 starting (1, 2, 9, 44, 265, ...)

A301423 Numbers k such that !k/(k-1) is prime, where !k = A000166(k) is the subfactorial of k.

Original entry on oeis.org

4, 5, 6, 11, 15, 44, 66, 168, 575, 1713

Offset: 1

Amiram Eldar, Mar 20 2018

Also numbers k such that A000255(k-2) is prime.
The corresponding primes are 3, 11, 53, 1468457, 34361893981, 22742406079421034331584846001936724930824184898296683, ...
a(11) > 35000. - Robert Price, Apr 14 2018

Cf. A000166, A000255.

Select[Range[2,100], PrimeQ[Subfactorial[#]/(#-1)] &]

isok(n) = (n != 1) && isprime(n!*sum(k=0, n, (-1)^k/k!)/(n-1)); \\ Michel Marcus, Mar 24 2018

A378160 The number of distinct prime factors of !n = A000166(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 2, 4, 3, 3, 4, 3, 3, 4, 5, 5, 4, 7, 5, 6, 4, 5, 7, 6, 6, 7, 4, 4, 4, 8, 4, 6, 4, 5, 6, 6, 4, 7, 2, 4, 7, 8, 6, 5, 7, 6, 7, 7, 4, 6, 9, 6, 6, 6, 6, 6, 4, 4, 5, 4, 3, 6, 6, 6, 6, 6, 7, 7, 4, 8, 6, 5, 8, 6, 4, 4, 5, 8, 4, 7, 7, 8, 6

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A000166, A001221, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378161, A378162.

Array[PrimeNu[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(omega(s), ", "));}

a(n) = A001221(A000166(n)).

a(n) >= A001221(n-1) + 1 for n >=5.

a(82)-a(88) from Jinyuan Wang, Nov 24 2024

A378161 The number of prime factors of !n = A000166(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 3, 6, 5, 3, 3, 6, 3, 3, 5, 6, 3, 8, 5, 6, 5, 8, 5, 9, 5, 5, 11, 7, 6, 9, 4, 8, 6, 8, 4, 10, 5, 5, 7, 8, 4, 8, 2, 7, 12, 8, 6, 9, 8, 7, 8, 8, 4, 10, 10, 8, 7, 6, 6, 8, 4, 4, 8, 9, 3, 8, 6, 7, 7, 6, 7, 13, 4, 8, 8, 6, 9, 7, 4, 7, 10, 8, 4, 9, 7

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A001222, A000166, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378160, A378162.

Array[PrimeOmega[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(bigomega(s), ", "));}

a(n) = A001222(A000166(n)).

a(n) >= A001222(n-1) + 1 for n >=4, with equality if and only if n is in A301423.

a(82)-a(86) from Jinyuan Wang, Nov 24 2024

cp's OEIS Frontend

A082491 a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).

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A101559 This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).

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A334715 A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Mathematica

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A378157 The least prime dividing !n = A000166(n).

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A065087 a(n) = A000166(n)*binomial(n+1,2).

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Mathematica

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A173184 Partial sums of A000166.

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A159610 Triangle read by rows, n-th row = n terms of A000255: (1, 3, 11, 53, 309, ...); right border = A000166 starting (1, 2, 9, 44, 265, ...).

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A301423 Numbers k such that !k/(k-1) is prime, where !k = A000166(k) is the subfactorial of k.

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