A009179 -id:A009179 - cp's OEIS frontend

A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 413496759611120779881, 9510425471055777937262

Offset: 0

N. J. A. Sloane

Euler (1809) not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n) = (n-1)*(a(n-1) + a(n-2)) and a(n) = n*a(n-1) + (-1)^n.
a(n) is the permanent of the matrix with 0 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003
a(n) is the number of desarrangements of length n. A desarrangement of length n is a permutation p of {1,2,...,n} for which the smallest of all the ascents of p (taken to be n if there are no ascents) is even. Example: a(3) = 2 because we have 213 and 312 (smallest ascents at i = 2). See the J. Désarménien link and the Bona reference (p. 118). - Emeric Deutsch, Dec 28 2007
a(n) is the number of deco polyominoes of height n and having in the last column an even number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. - Emeric Deutsch, Dec 28 2007
Attributed to Nicholas Bernoulli in connection with a probability problem that he presented. See Problem #15, p. 494, in "History of Mathematics" by David M. Burton, 6th edition. - Mohammad K. Azarian, Feb 25 2008
a(n) is the number of permutations p of {1,2,...,n} with p(1)!=1 and having no right-to-left minima in consecutive positions. Example a(3) = 2 because we have 231 and 321. - Emeric Deutsch, Mar 12 2008
a(n) is the number of permutations p of {1,2,...,n} with p(n)! = n and having no left to right maxima in consecutive positions. Example a(3) = 2 because we have 312 and 321. - Emeric Deutsch, Mar 12 2008
Number of wedged (n-1)-spheres in the homotopy type of the Boolean complex of the complete graph K_n. - Bridget Tenner, Jun 04 2008
The only prime number in the sequence is 2. - Howard Berman (howard_berman(AT)hotmail.com), Nov 08 2008
From Emeric Deutsch, Apr 02 2009: (Start)
a(n) is the number of permutations of {1,2,...,n} having exactly one small ascent. A small ascent in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. (Example: a(3) = 2 because we have 312 and 231; see the Charalambides reference, pp. 176-180.) [See also David, Kendall and Barton, p. 263. - N. J. A. Sloane, Apr 11 2014]
a(n) is the number of permutations of {1,2,...,n} having exactly one small descent. A small descent in a permutation (p_1,p_2,...,p_n) is a position i such that p_i - p_{i+1} = 1. (Example: a(3)=2 because we have 132 and 213.) (End)
For n > 2, a(n) + a(n-1) = A000255(n-1); where A000255 = (1, 1, 3, 11, 53, ...). - Gary W. Adamson, Apr 16 2009
Connection to A002469 (game of mousetrap with n cards): A002469(n) = (n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610.) - Gary W. Adamson, Apr 17 2009
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) is the sum of the values of the largest fixed points of all non-derangements of length n-1. Example: a(4)=9 because the non-derangements of length 3 are 123, 132, 213, and 321, having largest fixed points 3, 1, 3, and 2, respectively.
a(n) is the number of non-derangements of length n+1 for which the difference between the largest and smallest fixed point is 2. Example: a(3) = 2 because we have 1'43'2 and 32'14'; a(4) = 9 because we have 1'23'54, 1'43'52, 1'53'24, 52'34'1, 52'14'3, 32'54'1, 213'45', 243'15', and 413'25' (the extreme fixed points are marked).
(End)
a(n), n >= 1, is also the number of unordered necklaces with n beads, labeled differently from 1 to n, where each necklace has >= 2 beads. This produces the M2 multinomial formula involving partitions without part 1 given below. Because M2(p) counts the permutations with cycle structure given by partition p, this formula gives the number of permutations without fixed points (no 1-cycles), i.e., the derangements, hence the subfactorials with their recurrence relation and inputs. Each necklace with no beads is assumed to contribute a factor 1 in the counting, hence a(0)=1. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 01 2010
From Emeric Deutsch, Sep 06 2010: (Start)
a(n) is the number of permutations of {1,2,...,n, n+1} starting with 1 and having no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 1324 and 1432.
a(n) is the number of permutations of {1,2,...,n} that do not start with 1 and have no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 213 and 321.
(End)
Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleave except on the leftmost path, there is no vertex of outdegree one on the leftmost path. - Wenjin Woan, May 23 2011
a(n) is the number of zeros in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 2 pure options. - Raimundas Vidunas, Jan 22 2014
Convolution of sequence A135799 with the sequence generated by 1+x^2/(2*x+1). - Thomas Baruchel, Jan 08 2016
The number of interior lattice points of the subpolytope of the n-dimensional permutohedron whose vertices correspond to permutations avoiding 132 and 312. - Robert Davis, Oct 05 2016
Consider n circles of different radii, where each circle is either put inside some bigger circle or contains a smaller circle inside it (no common points are allowed). Then a(n) gives the number of such combinations. - Anton Zakharov, Oct 12 2016
If we partition the permutations of [n+1] in A000240 according to their starting digit, we will get (n+1) equinumerous classes each of size a(n), i.e., A000240(n+1) = (n+1)*a(n), hence a(n) is the size of each class of permutations of [n+1] in A000240. For example, for n = 4 we have 45 = 5*9. - Enrique Navarrete, Jan 10 2017
Call d_n1 the permutations of [n] that have the substring n1 but no substring in {12,23,...,(n-1)n}. If we partition them according to their starting digit, we will get (n-1) equinumerous classes each of size A000166(n-2) (the class starting with the digit 1 is empty since we must have the substring n1). Hence d_n1 = (n-1)*A000166(n-2) and A000166(n-2) is the size of each nonempty class in d_n1. For example, d_71 = 6*44 = 264, so there are 264 permutations in d_71 distributed in 6 nonempty classes of size A000166(5) = 44. (To get permutations in d_n1 recursively from more basic ones see the link "Forbidden Patterns" below.) - Enrique Navarrete, Jan 15 2017
Also the number of maximum matchings and minimum edge covers in the n-crown graph. - Eric W. Weisstein, Jun 14 and Dec 24 2017
The sequence a(n) taken modulo a positive integer k is periodic with exact period dividing k when k is even and dividing 2*k when k is odd. This follows from the congruence a(n+k) = (-1)^k*a(n) (mod k) for all n and k, which in turn is easily proved by induction making use of the recurrence a(n) = n*a(n-1) + (-1)^n. - Peter Bala, Nov 21 2017
a(n) is the number of distinct possible solutions for a directed, no self loop containing graph (not necessarily connected) that has n vertices, and each vertex has an in- and out-degree of exactly 1. - Patrik Holopainen, Sep 18 2018
a(n) is the dimension of the kernel of the random-to-top and random-to-random shuffling operators over a collection of n objects (in a vector space of size n!), as noticed by M. Wachs and V. Reiner. See the Reiner, Saliola and Welker reference below. - Nadia Lafreniere, Jul 18 2019
a(n) is the number of distinct permutations for a Secret Santa gift exchange with n participants. - Patrik Holopainen, Dec 30 2019
a(2*n+1) is even. More generally, a(m*n+1) is divisible by m*n, which follows from a(n+1) = n*(a(n) + a(n-1)) = n*A000255(n-1) for n >= 1. a(2*n) is odd; in fact, a(2*n) == 1 (mod 8). Other divisibility properties include a(6*n) == 1 (mod 24), a(9*n+4) == a(9*n+7) == 0 (mod 9), a(10*n) == 1 (mod 40), a(11*n+5) == 0 (mod 11) and a(13*n+8 ) == 0 (mod 13). - Peter Bala, Apr 05 2022
Conjecture: a(n) with n > 2 is a perfect power only for n = 4 with a(4) = 3^2. This has been verified for n <= 1000. - Zhi-Wei Sun, Jan 09 2025

			a(2) = 1, a(3) = 2 and a(4) = 9 since the possibilities are {BA}, {BCA, CAB} and {BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA}. - _Henry Bottomley_, Jan 17 2001
The Boolean complex of the complete graph K_4 is homotopy equivalent to the wedge of 9 3-spheres.
Necklace problem for n = 6: partitions without part 1 and M2 numbers for n = 6: there are A002865(6) = 4 such partitions, namely (6), (2,4), (3^2) and (2^3) in A-St order with the M2 numbers 5!, 90, 40 and 15, respectively, adding up to 265 = a(6). This corresponds to 1 necklace with 6 beads, two necklaces with 2 and 4 beads respectively, two necklaces with 3 beads each and three necklaces with 2 beads each. - _Wolfdieter Lang_, Jun 01 2010
G.f. = 1 + x^2 + 9*x^3 + 44*x^4 + 265*x^5 + 1854*x^6 + 14833*x^7 + 133496*x^8 + ...

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Index entries for "core" sequences.
Index entries for sequences related to binary matrices.

Cf. A000142, A002467, A003221, A000522, A000240, A000387, A000449, A000475, A129135, A092582, A000255, A002469, A159610, A068985, A068996, A047865, A038205, A008279, A281682.
For the probabilities a(n)/n!, see A053557/A053556 and A103816/A053556.
A diagonal of A008291 and A068106. Column A008290(n,0).
A001120 has a similar recurrence.
For other derangement numbers see also A053871, A033030, A088991, A088992.
Pairwise sums of A002741 and A000757. Differences of A001277.
Cf. A101560, A101559, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.
A diagonal in triangles A008305 and A010027.
a(n)/n! = A053557/A053556 = (N(n, n) of A103361)/(D(n, n) of A103360).
Column k=0 of A086764 and of A334715. Column k=1 of A364068.
Row sums of A216963 and of A323671.

a000166 n = a000166_list !! n
a000166_list = 1 : 0 : zipWith (*) [1..]
                       (zipWith (+) a000166_list $ tail a000166_list)
-- Reinhard Zumkeller, Dec 09 2012

I:=[0,1]; [1] cat [n le 2 select I[n] else (n-1)*(Self(n-1)+Self(n-2)): n in [1..30]]; // Vincenzo Librandi, Jan 07 2016

A000166 := proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;
a:=n->n!*sum((-1)^k/k!, k=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, May 17 2007
ZL1:=[S,{S=Set(Cycle(Z,card>1))},labeled]: seq(count(ZL1,size=n),n=0..21); # Zerinvary Lajos, Sep 26 2007
with (combstruct):a:=proc(m) [ZL,{ZL=Set(Cycle(Z,card>=m))},labeled]; end: A000166:=a(2):seq(count(A000166,size=n),n=0..21); # Zerinvary Lajos, Oct 02 2007
Z := (x, m)->m!^2*sum(x^j/((m-j)!^2), j=0..m): R := (x, n, m)->Z(x, m)^n: f := (t, n, m)->sum(coeff(R(x, n, m), x, j)*(t-1)^j*(n*m-j)!, j=0..n*m): seq(f(0, n, 1), n=0..21); # Zerinvary Lajos, Jan 22 2008
a:=proc(n) if `mod`(n,2)=1 then sum(2*k*factorial(n)/factorial(2*k+1), k=1.. floor((1/2)*n)) else 1+sum(2*k*factorial(n)/factorial(2*k+1), k=1..floor((1/2)*n)-1) end if end proc: seq(a(n),n=0..20); # Emeric Deutsch, Feb 23 2008
G(x):=2*exp(-x)/(1-x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/2,n=0..21); # Zerinvary Lajos, Apr 03 2009
seq(simplify(KummerU(-n, -n, -1)), n = 0..23); # Peter Luschny, May 10 2022

a[0] = 1; a[n_] := n*a[n - 1] + (-1)^n; a /@ Range[0, 21] (* Robert G. Wilson v *)
a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 (* Michael Taktikos, May 26 2006 *)
Range[0, 20]! CoefficientList[ Series[ Exp[ -x]/(1 - x), {x, 0, 20}], x]
dr[{n_,a1_,a2_}]:={n+1,a2,n(a1+a2)}; Transpose[NestList[dr,{0,0,1},30]][[3]] (* Harvey P. Dale, Feb 23 2013 *)
a[n_] := (-1)^n HypergeometricPFQ[{- n, 1}, {}, 1]; (* Michael Somos, Jun 01 2013 *)
a[n_] := n! SeriesCoefficient[Exp[-x] /(1 - x), {x, 0, n}]; (* Michael Somos, Jun 01 2013 *)
Table[Subfactorial[n], {n, 0, 21}] (* Jean-François Alcover, Jan 10 2014 *)
RecurrenceTable[{a[n] == n*a[n - 1] + (-1)^n, a[0] == 1}, a, {n, 0, 23}] (* Ray Chandler, Jul 30 2015 *)
Subfactorial[Range[0, 20]] (* Eric W. Weisstein, Dec 31 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1)}; NestList[nxt,{0,1},25][[All,2]] (* Harvey P. Dale, Jun 01 2019 *)

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

{a(n) = if( n<1, 1, n * a(n-1) + (-1)^n)}; /* Michael Somos, Mar 24 2003 */

{a(n) = n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}; /* Michael Somos, Mar 24 2003 */

{a(n)=polcoeff(sum(m=0,n,m^m*x^m/(1+(m+1)*x+x*O(x^n))^(m+1)),n)} /* Paul D. Hanna */

A000166=n->n!*sum(k=0,n,(-1)^k/k!) \\ M. F. Hasler, Jan 26 2012

a(n)=if(n,round(n!/exp(1)),1) \\ Charles R Greathouse IV, Jun 17 2012

apply( {A000166(n)=n!\/exp(n>0)}, [0..22]) \\ M. F. Hasler, Nov 09 2024

Python
```
See Hobson link.
```

A000166_list, m, x = [], 1, 1
for n in range(10*2):
    x, m = x*n + m, -m
    A000166_list.append(x) # Chai Wah Wu, Nov 03 2014

(A000166 + A000522)/2 = A009179, (A000166 - A000522)/2 = A009628.
a(n) = A008290(n,0).
a(n) + A003048(n+1) = 2*n!. - D. G. Rogers, Aug 26 2006
a(n) = {(n-1)!/exp(1)}, n > 1, where {x} is the nearest integer function. - Simon Plouffe, March 1993 [This uses offset 1, see below for the version with offset 0. - Charles R Greathouse IV, Jan 25 2012]
a(0) = 1, a(n) = round(n!/e) = floor(n!/e + 1/2) for n > 0.
a(n) = n!*Sum_{k=0..n} (-1)^k/k!.
D-finite with recurrence a(n) = (n-1)*(a(n-1) + a(n-2)), n > 0.
a(n) = n*a(n-1) + (-1)^n.
E.g.f.: exp(-x)/(1-x).
a(n) = Sum_{k=0..n} binomial(n, k)*(-1)^(n-k)*k! = Sum_{k=0..n} (-1)^(n-k)*n!/(n-k)!. - Paul Barry, Aug 26 2004
The e.g.f. y(x) satisfies y' = x*y/(1-x).
Inverse binomial transform of A000142. - Ross La Haye, Sep 21 2004
In Maple notation, representation as n-th moment of a positive function on [-1, infinity]: a(n)= int( x^n*exp(-x-1), x=-1..infinity ), n=0, 1... . a(n) is the Hamburger moment of the function exp(-1-x)*Heaviside(x+1). - Karol A. Penson, Jan 21 2005
a(n) = A001120(n) - n!. - Philippe Deléham, Sep 04 2005
a(n) = Integral_{x=0..oo} (x-1)^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006
a(n) = Sum_{k=2,4,...} T(n,k), where T(n,k) = A092582(n,k) = k*n!/(k+1)! for 1 <= k < n and T(n,n)=1. - Emeric Deutsch, Feb 23 2008
a(n) = n!/e + (-1)^n*(1/(n+2 - 1/(n+3 - 2/(n+4 - 3/(n+5 - ...))))). Asymptotic result (Ramanujan): (-1)^n*(a(n) - n!/e) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + ..., where the sequence [1,2,5,15,...] is the sequence of Bell numbers A000110. - Peter Bala, Jul 14 2008
From William Vaughn (wvaughn(AT)cvs.rochester.edu), Apr 13 2009: (Start)
a(n) = Integral_{p=0..1} (log(1/(1-p)) - 1)^n dp.
Proof: Using the substitutions 1=log(e) and y = e(1-p) the above integral can be converted to ((-1)^n/e) Integral_{y=0..e} (log(y))^n dy.
From CRC Integral tables we find the antiderivative of (log(y))^n is (-1)^n n! Sum_{k=0..n} (-1)^k y(log(y))^k / k!.
Using the fact that e(log(e))^r = e for any r >= 0 and 0(log(0))^r = 0 for any r >= 0 the integral becomes n! * Sum_{k=0..n} (-1)^k / k!, which is line 9 of the Formula section. (End)
a(n) = exp(-1)*Gamma(n+1,-1) (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009
G.f.: 1/(1-x^2/(1-2x-4x^2/(1-4x-9x^2/(1-6x-16x^2/(1-8x-25x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009
a(n) = Sum_{p in Pano1(n)} M2(p), n >= 1, with Pano1(n) the set of partitions without part 1, and the multinomial M2 numbers. See the characteristic array for partitions without part 1 given by A145573 in Abramowitz-Stegun (A-S) order, with A002865(n) the total number of such partitions. The M2 numbers are given for each partition in A-St order by the array A036039. - Wolfdieter Lang, Jun 01 2010
a(n) = row sum of A008306(n), n > 1. - Gary Detlefs, Jul 14 2010
a(n) = ((-1)^n)*(n-1)*hypergeom([-n+2, 2], [], 1), n>=1; 1 for n=0. - Wolfdieter Lang, Aug 16 2010
a(n) = (-1)^n * hypergeom([ -n, 1], [], 1), n>=1; 1 for n=0. From the binomial convolution due to the e.g.f. - Wolfdieter Lang, Aug 26 2010
Integral_{x=0..1} x^n*exp(x) = (-1)^n*(a(n)*e - n!).
O.g.f.: Sum_{n>=0} n^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Oct 06 2011
Abs((a(n) + a(n-1))*e - (A000142(n) + A000142(n-1))) < 2/n. - Seiichi Kirikami, Oct 17 2011
G.f.: hypergeom([1,1],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
From Sergei N. Gladkovskii, Nov 25 2011, Jul 05 2012, Sep 23 2012, Oct 13 2012, Mar 09 2013, Mar 10 2013, Oct 18 2013: (Start)
Continued fractions:
In general, e.g.f. (1+a*x)/exp(b*x) = U(0) with U(k) = 1 + a*x/(1-b/(b-a*(k+1)/U(k+1))). For a=-1, b=-1: exp(-x)/(1-x) = 1/U(0).
E.g.f.: (1-x/(U(0)+x))/(1-x), where U(k) = k+1 - x + (k+1)*x/U(k+1).
E.g.f.: 1/Q(0) where Q(k) = 1 - x/(1 - 1/(1 - (k+1)/Q(k+1))).
G.f.: 1/U(0) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: Q(0)/(1+x) where Q(k) = 1 + (2*k+1)*x/((1+x)-2*x*(1+x)*(k+1)/(2*x*(k+1)+(1+x)/ Q(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2-(1-2*x*k)*(1-2*x-2*x*k)/T(k+1)). (End)
0 = a(n)*(a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n>=0. - Michael Somos, Jan 25 2014
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^k*(k + x + 1)^(n-k) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^(n-k)*(k + x - 1)^k, for arbitrary x. - Peter Bala, Feb 19 2017
From Peter Luschny, Jun 20 2017: (Start)
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(-j-1, -n-1)*abs(Stirling1(j, k)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Pochhammer(n-k+1, k) (cf. A008279). (End)
a(n) = n! - Sum_{j=0..n-1} binomial(n,j) * a(j). - Alois P. Heinz, Jan 23 2019
Sum_{n>=2} 1/a(n) = A281682. - Amiram Eldar, Nov 09 2020
a(n) = KummerU(-n, -n, -1). - Peter Luschny, May 10 2022
a(n) = (-1)^n*Sum_{k=0..n} Bell(k)*Stirling1(n+1, k+1). - Mélika Tebni, Jul 05 2022

Minor edits by M. F. Hasler, Jan 16 2017

A000522 Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

Original entry on oeis.org

1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101, 108505112, 1302061345, 16926797486, 236975164805, 3554627472076, 56874039553217, 966858672404690, 17403456103284421, 330665665962404000, 6613313319248080001, 138879579704209680022, 3055350753492612960485, 70273067330330098091156

Offset: 0

N. J. A. Sloane

Total number of permutations of all subsets of an n-set.
Also the number of one-to-one sequences that can be formed from n distinct objects.
Old name "Total number of permutations of a set with n elements", or the same with the word "arrangements", both sound too much like A000142.
Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.
a(n) is also the number of paths (without loops) in the complete graph on n+2 vertices starting at one vertex v1 and ending at another v2. Example: when n=2 there are 5 paths in the complete graph with 4 vertices starting at the vertex 1 and ending at the vertex 2: (12),(132),(142),(1342),(1432) so a(2) = 5. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001; comment corrected by Jonathan Coxhead, Mar 21 2003
Also row sums of Table A008279, which can be generated by taking the derivatives of x^k. For example, for y = 1*x^3, y' = 3x^2, y" = 6x, y'''= 6 so a(4) = 1 + 3 + 6 + 6 = 16. - Alford Arnold, Dec 15 1999
a(n) is the permanent of the n X n matrix with 2s on the diagonal and 1s elsewhere. - Yuval Dekel, Nov 01 2003
(A000166 + this_sequence)/2 = A009179, (A000166 - this_sequence)/2 = A009628.
Stirling transform of A006252(n-1) = [1,1,1,2,4,14,38,...] is a(n-1) = [1,2,5,16,65,...]. - Michael Somos, Mar 04 2004
Number of {12,12*,21*}- and {12,12*,2*1}-avoiding signed permutations in the hyperoctahedral group.
a(n) = b such that Integral_{x=0..1} x^n*exp(-x) dx = a-b*exp(-1). - Sébastien Dumortier, Mar 05 2005
a(n) is the number of permutations on [n+1] whose left-to-right record lows all occur at the start. Example: a(2) counts all permutations on [3] except 231 (the last entry is a record low but its predecessor is not). - David Callan, Jul 20 2005
a(n) is the number of permutations on [n+1] that avoid the (scattered) pattern 1-2-3|. The vertical bar means the "3" must occur at the end of the permutation. For example, 21354 is not counted by a(4): 234 is an offending subpermutation. - David Callan, Nov 02 2005
Number of deco polyominoes of height n+1 having no reentrant corners along the lower contour (i.e., no vertical step that is followed by a horizontal step). In other words, a(n)=A121579(n+1,0). 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(1)=2 because the only deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along their lower contours. - Emeric Deutsch, Aug 16 2006
Unreduced numerators of partial sums of the Taylor series for e. - Jonathan Sondow, Aug 18 2006
a(n) is the number of permutations on [n+1] (written in one-line notation) for which the subsequence beginning at 1 is increasing. Example: a(2)=5 counts 123, 213, 231, 312, 321. - David Callan, Oct 06 2006
a(n) is the number of permutations (written in one-line notation) on the set [n + k], k >= 1, for which the subsequence beginning at 1,2,...,k is increasing. Example: n = 2, k = 2. a(2) = 5 counts 1234, 3124, 3412, 4123, 4312. - Peter Bala, Jul 29 2014
a(n) and (1,-2,3,-4,5,-6,7,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Nov 01 2007
Consider the subsets of the set {1,2,3,...,n} formed by the first n integers. E.g., for n = 3 we have {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all possible subsets is written as Sum_{sbst=subsets}. Then a(n) = Sum_{sbst=subsets} nprts(sbst)!. E.g., for n = 3 we have 1!+1!+1!+1!+2!+2!+2!+3!=16. - Thomas Wieder, Jun 17 2006
Equals row sums of triangle A158359(unsigned). - Gary W. Adamson, Mar 17 2009
Equals eigensequence of triangle A158821. - Gary W. Adamson, Mar 30 2009
For positive n, equals 1/BarnesG(n+1) times the determinant of the n X n matrix whose (i,j)-coefficient is the (i+j)th Bell number. - John M. Campbell, Oct 03 2011
a(n) is the number of n X n binary matrices with i) at most one 1 in each row and column and ii) the subset of rows that contain a 1 must also be the columns that contain a 1. Cf. A002720 where restriction ii is removed. - Geoffrey Critzer, Dec 20 2011
Number of restricted growth strings (RGS) [d(1),d(2),...,d(n)] such that d(k) <= k and d(k) <= 1 + (number of nonzero digits in prefix). The positions of nonzero digits determine the subset, and their values (decreased by 1) are the (left) inversion table (a rising factorial number) for the permutation, see example. - Joerg Arndt, Dec 09 2012
Number of a restricted growth strings (RGS) [d(0), d(1), d(2), ..., d(n)] where d(k) >= 0 and d(k) <= 1 + chg([d(0), d(1), d(2), ..., d(k-1)]) and chg(.) gives the number of changes of its argument. Replacing the function chg(.) by a function asc(.) that counts the ascents in the prefix gives A022493 (ascent sequences). - Joerg Arndt, May 10 2013
The sequence t(n) = number of i <= n such that floor(e*i!) is a square is mentioned in the abstract of Luca & Shparlinski. The values are t(n) = 0 for 0 <= n <= 2 and t(n) = 1 for at least 3 <= n <= 300. - R. J. Mathar, Jan 16 2014
a(n) = p(n,1) = q(n,1), where p and q are polynomials defined at A248664 and A248669. - Clark Kimberling, Oct 11 2014
a(n) is the number of ways at most n people can queue up at a (slow) ticket counter when one or more of the people may choose not to queue up. Note that there are C(n,k) sets of k people who quene up and k! ways to queue up. Since k can run from 0 to n, a(n) = Sum_{k=0..n} n!/(n-k)! = Sum_{k=0..n} n!/k!. For example, if n=3 and the people are A(dam), B(eth), and C(arl), a(3)=16 since there are 16 possible lineups: ABC, ACB, BAC, BCA, CAB, CBA, AB, BA, AC, CA, BC, CB, A, B, C, and empty queue. - Dennis P. Walsh, Oct 02 2015
As the row sums of A008279, Motzkin uses the abbreviated notation $n_<^\Sigma$ for a(n).
The piecewise polynomial function f defined by f(x) = a(n)*x^n/n! on each interval [ 1-1/a(n), 1-1/a(n+1) ) is continuous on [0,1) and lim_{x->1} f(x) = e. - Luc Rousseau, Oct 15 2019
a(n) is composite for 3 <= n <= 2015, but a(2016) is prime (or at least a strong pseudoprime): see Johansson link. - Robert Israel, Jul 27 2020 [a(2016) is prime, ECPP certificate generated with CM 0.4.3 and checked by factordb. - Jason H Parker, Jun 15 2025]
In general, sequences of the form a(0)=a, a(n) = n*a(n-1) + k, n>0, will have a closed form of n!*a + floor(n!*(e-1))*k. - Gary Detlefs, Oct 26 2020
From Peter Bala, Apr 03 2022: (Start)
a(2*n) is odd and a(2*n+1) is even. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(5*n+2) == a(5*n+4) == 0 (mod 5), a(25*n+7) == a(25*n+19) == 0 (mod 25) and a(13*n+4) == a(13*n+10)== a(13*n+12) == 0 (mod 13). (End)
Number of possible ranking options on a typical ranked choice voting ballot with n candidates (allowing undervotes). - P. Christopher Staecker, May 05 2024
From Thomas Scheuerle, Dec 28 2024: (Start)
Number of decorated permutations of size n.
Number of Le-diagrams with bounding box semiperimeter n, for n > 0.
By counting over all k = 1..n and n > 0, the number of positroid cells for the totally nonnegative real Grassmannian Gr(k, n), equivalently the number of Grassmann necklaces of type (k, n). (End)

			G.f. = 1 + 2*x + 5*x^2 + 16*x^3 + 65*x^4 + 326*x^5 + 1957*x^6 + 13700*x^7 + ...
With two objects we can form 5 sequences: (), (a), (b), (a,b), (b,a), so a(2) = 5.
From _Joerg Arndt_, Dec 09 2012: (Start)
The 16 arrangements of the 3-set and their RGS (dots denote zeros) are
  [ #]       RGS        perm. of subset
  [ 1]    [ . . . ]      [  ]
  [ 2]    [ . . 1 ]      [ 3 ]
  [ 3]    [ . 1 . ]      [ 2 ]
  [ 4]    [ . 1 1 ]      [ 2 3 ]
  [ 5]    [ . 1 2 ]      [ 3 2 ]
  [ 6]    [ 1 . . ]      [ 1 ]
  [ 7]    [ 1 . 1 ]      [ 1 3 ]
  [ 8]    [ 1 . 2 ]      [ 3 1 ]
  [ 9]    [ 1 1 . ]      [ 1 2 ]
  [10]    [ 1 1 1 ]      [ 1 2 3 ]
  [11]    [ 1 1 2 ]      [ 1 3 2 ]
  [12]    [ 1 1 3 ]      [ 2 3 1 ]
  [13]    [ 1 2 . ]      [ 2 1 ]
  [14]    [ 1 2 1 ]      [ 2 1 3 ]
  [15]    [ 1 2 2 ]      [ 3 1 2 ]
  [16]    [ 1 2 3 ]      [ 3 2 1 ]
(End)

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CombOS - Combinatorial Object Server, Generate partial permutations
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P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 114.
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Bernd Gaertner, Walter D. Jr. Morris and Leo Ruest, Unique Sink Orientations of Grids, Algorithmica, Volume 51, Number 2 / June 2008.
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Eric Weisstein's World of Mathematics, Binomial Sums
Index entries for sequences related to logarithmic numbers
Index entries for related partition-counting sequences
Index entries for sequences related to factorial numbers

Cf. A000166, A002627, A006231, A064383, A064384, A008290, A010844, A010845, A014508, A038159, A054091, A058006, A072453, A072456, A073591, A082030, A095000, A095177, A108625, A121579, A124779, A142992, A143007, A158359, A158821, A195254, A222637-A222639, A038155, A000217.
Average of n-th row of triangle in A068424 [Corrected by N. J. A. Sloane, Feb 29 2024].
Row sums of A008279 and A094816.
First differences give A001339. Second differences give A001340.
Partial sums are in A001338, A002104.
A row of the array in A144502.
See also A370973, Nearest integer to e*n!.

import Data.List (subsequences, permutations)
a000522 = length . choices . enumFromTo 1 where
choices = concat . map permutations . subsequences
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

[1] cat [n eq 1 select (n+1) else n*Self(n-1)+1: n in [1..25]]; // Vincenzo Librandi, Feb 15 2015

a(n):= exp(1)*int(x^n*exp(-x)*Heaviside(x-1), x=0..infinity); # Karol A. Penson, Oct 01 2001
A000522 := n->add(n!/k!,k=0..n);
G(x):=exp(x)/(1-x): 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=0..20);
# Zerinvary Lajos, Apr 03 2009
G:=exp(z)/(1-z): Gser:=series(G,z=0,21):
for n from 0 to 20 do a(n):=n!*coeff(Gser,z,n): end do
# Paul Weisenhorn, May 30 2010
k := 1; series(hypergeom([1,k],[],x/(1-x))/(1-x), x=0, 20); # Mark van Hoeij, Nov 07 2011
# one more Maple program:
a:= proc(n) option remember;
      `if`(n<0, 0, 1+n*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 13 2019
seq(simplify(KummerU(-n, -n, 1)), n = 0..23); # Peter Luschny, May 10 2022

Table[FunctionExpand[Gamma[n + 1, 1]*E], {n, 0, 24}]
nn = 20; Accumulate[Table[1/k!, {k, 0, nn}]] Range[0, nn]! (* Jan Mangaldan, Apr 21 2013 *)
FoldList[#1*#2 + #2 &, 0, Range@ 23] + 1 (* or *)
f[n_] := Floor[E*n!]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Feb 13 2015 *)
RecurrenceTable[{a[n + 1] == (n + 1) a[n] + 1, a[0] == 1}, a, {n, 0, 12}] (* Emanuele Munarini, Apr 27 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+1}; NestList[nxt,{0,1},30][[All,2]] (* Harvey P. Dale, Jan 29 2023 *)

a(n) := if n=0 then 1 else n*a(n-1)+1; makelist(a(n),n,0,12); /* Emanuele Munarini, Apr 27 2017 */

{a(n) = local(A); if( n<0, 0, A = vector(n+1); A[1]=1; for(k=1, n, A[k+1] = k*A[k] + 1); A[n+1])}; /* Michael Somos, Jul 01 2004 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x +x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Mar 06 2004 */

a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x)^2+x^2*deriv(A)/(1-x)); polcoeff(A, n) \\ Paul D. Hanna, Sep 03 2008

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,(m+2)^m*x^m/(1+(m+1)*X)^(m+1)),n)} /* Paul D. Hanna */

a(n)=sum(k=0,n,binomial(n,k)*k!); \\ Joerg Arndt, Dec 14 2014

# program adapted from Alois P. Heinz's Maple code in A022493
@CachedFunction
def b(n, i, t):
    if n <= 1:
        return 1
    return sum(b(n - 1, j, t + (j == i)) for j in range(t + 2))
def a(n):
    return b(n, 0, 0)
v000522 = [a(n) for n in range(33)]
print(v000522)
# Joerg Arndt, May 11 2013

a(n) = n*a(n-1) + 1, a(0) = 1.
a(n) = A007526(n-1) + 1.
a(n) = A061354(n)*A093101(n).
a(n) = n! * Sum_{k=0..n} 1/k! = n! * (e - Sum_{k>=n+1} 1/k!). - Michael Somos, Mar 26 1999
a(0) = 1; for n > 0, a(n) = floor(e*n!).
E.g.f.: exp(x)/(1-x).
a(n) = 1 + Sum_{n>=k>=j>=0} (k-j+1)*k!/j! = a(n-1) + A001339(n-1) = A007526(n) + 1. Binomial transformation of n!, i.e., A000142. - Henry Bottomley, Jun 04 2001
a(n) = floor(2/(n+1))*A009578(n+1)-1. - Emeric Deutsch, Oct 24 2001
Integral representation as n-th moment of a nonnegative function on a positive half-axis: a(n) = e*Integral_{x>=0} x^n*e^(-x)*Heaviside(x-1) dx. - Karol A. Penson, Oct 01 2001
Formula, in Mathematica notation: Special values of Laguerre polynomials, a(n)=(-1)^n*n!*LaguerreL[n, -1-n, 1], n=1, 2, ... . This relation cannot be checked by Maple, as it appears that Maple does not incorporate Laguerre polynomials with second index equal to negative integers. It does check with Mathematica. - Karol A. Penson and Pawel Blasiak ( blasiak(AT)lptl.jussieu.fr), Feb 13 2004
G.f.: Sum_{k>=0} k!*x^k/(1-x)^(k+1). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*k^(n-k)*(k+1)^k. - Vladeta Jovovic, Aug 18 2002
a(n) = Sum_{k=0..n} A008290(n, k)*2^k. - Philippe Deléham, Dec 12 2003
a(n) = Sum_{k=0..n} A046716(n, k). - Philippe Deléham, Jun 12 2004
a(n) = e*Gamma(n+1,1) where Gamma(z,t) = Integral_{x>=t} e^(-x)*x^(z-1) dx is incomplete gamma function. - Michael Somos, Jul 01 2004
a(n) = Sum_{k=0..n} P(n, k). - Ross La Haye, Aug 28 2005
Given g.f. A(x), then g.f. A059115 = A(x/(1-x)). - Michael Somos, Aug 03 2006
a(n) = 1 + n + n*(n-1) + n*(n-1)*(n-2) + ... + n!. - Jonathan Sondow, Aug 18 2006
a(n) = Sum_{k=0..n} binomial(n,k) * k!; interpretation: for all k-subsets (sum), choose a subset (binomial(n,k)), and permutation of subset (k!). - Joerg Arndt, Dec 09 2012
a(n) = Integral_{x>=0} (x+1)^n*e^(-x) dx. - Gerald McGarvey, Oct 19 2006
a(n) = Sum_{k=0..n} A094816(n, k), n>=0 (row sums of Poisson-Charlier coefficient matrix). - N. J. A. Sloane, Nov 10 2007
From Tom Copeland, Nov 01 2007, Dec 10 2007: (Start)
Act on 1/(1-x) with 1/(1-xDx) = Sum_{j>=0} (xDx)^j = Sum_{j>=0} x^j*D^j*x^j = Sum_{j>=0} j!*x^j*L(j,-:xD:,0) where Lag(n,x,0) are the Laguerre polynomials of order 0, D the derivative w.r.t. x and (:xD:)^j = x^j*D^j. Truncating the operator series at the j = n term gives an o.g.f. for a(0) through a(n) consistent with Jovovic's.
These results and those of Penson and Blasiak, Arnold, Bottomley and Deleham are related by the operator A094587 (the reverse of A008279), which is the umbral equivalent of n!*Lag[n,(.)!*Lag[.,x,-1],0] = (1-D)^(-1) x^n = (-1)^n * n! Lag(n,x,-1-n) = Sum_{j=0..n} binomial(n,j)*j!*x^(n-j) = Sum_{j=0..n} (n!/j!)*x^j. Umbral substitution of b(.) for x and then letting b(n)=1 for all n connects the results. See A132013 (the inverse of A094587) for a connection between these operations and 1/(1-xDx).
(End)
From Peter Bala, Jul 15 2008: (Start)
a(n) = n!*e - 1/(n + 1/(n+1 + 2/(n+2 + 3/(n+3 + ...)))).
Asymptotic result (Ramanujan): n!*e - a(n) ~ 1/n - 1/n^3 + 1/n^4 + 2/n^5 - 9/n^6 + ..., where the sequence [1,0,-1,1,2,-9,...] = [(-1)^k*A000587(k)], for k>=1.
a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). For fixed k, define the derived sequence a_k(n) = (a(n+k)-a(k))/n, n = 1,2,3,... . Then a_k(n) is also a difference divisibility sequence.
For example, the derived sequence a_0(n) is just a(n-1). The set of integer sequences satisfying the difference divisibility property forms a ring with term-wise operations of addition and multiplication.
Recurrence relations: a(0) = 1, a(n) = (n-1)*(a(n-1) + a(n-2)) + 2, for n >= 1. a(0) = 1, a(1) = 2, D-finite with recurrence: a(n) = (n+1)*a(n-1) - (n-1)*a(n-2) for n >= 2. The sequence b(n) := n! satisfies the latter recurrence with the initial conditions b(0) = 1, b(1) = 1. This leads to the finite continued fraction expansion a(n)/n! = 1/(1-1/(2-1/(3-2/(4-...-(n-1)/(n+1))))), n >= 2.
Limit_{n->oo} a(n)/n! = e = 1/(1-1/(2-1/(3-2/(4-...-n/((n+2)-...))))). This is the particular case m = 0 of the general result m!/e - d_m = (-1)^(m+1) *(1/(m+2 -1/(m+3 -2/(m+4 -3/(m+5 -...))))), where d_m denotes the m-th derangement number A000166(m).
For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A001339 (r=1), A082030 (r=2), A095000 (r=3) and A095177 (r=4).
For the corresponding results for the constants log(2), zeta(2) and zeta(3) refer to A142992, A108625 and A143007 respectively.
(End)
G.f. satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x)/(1-x). - Paul D. Hanna, Sep 03 2008
From Paul Barry, Nov 27 2009: (Start)
G.f.: 1/(1-2*x-x^2/(1-4*x-4*x^2/(1-6*x-9*x^2/(1-8*x-16*x^2/(1-10*x-25*x^2/(1-... (continued fraction);
G.f.: 1/(1-x-x/(1-x/(1-x-2*x/(1-2*x/(1-x-3*x/(1-3*x/(1-x-4*x/(1-4*x/(1-x-5*x/(1-5*x/(1-... (continued fraction).
(End)
O.g.f.: Sum_{n>=0} (n+2)^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Sep 19 2011
G.f. hypergeom([1,k],[],x/(1-x))/(1-x), for k=1,2,...,9 is the generating function for A000522, A001339, A082030, A095000, A095177, A096307, A096341, A095722, and A095740. - Mark van Hoeij, Nov 07 2011
G.f.: 1/U(0) where U(k) = 1 - x - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2012
E.g.f.: 1/U(0) where U(k) = 1 - x/(1 - 1/(1 + (k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 16 2012
G.f.: 1/(1-x)/Q(0), where Q(k) = 1 - x/(1-x)*(k+1)/(1 - x/(1-x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
G.f.: 2/(1-x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+3) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: (B(x)+ 1)/(2-2*x) = Q(0)/(2-2*x), where B(x) be g.f. A006183, Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(k+1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
E.g.f.: e^x/(1-x) = (1 - 12*x/(Q(0) + 6*x - 3*x^2))/(1-x), where Q(k) = 2*(4*k+1)*(32*k^2 + 16*k + x^2 - 6) - x^4*(4*k-1)*(4*k+7)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
G.f.: conjecture: T(0)/(1-2*x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
0 = a(n)*(+a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(+a(n+1) - a(n+3)) + a(n+2)*(+a(n+2)) for all n>=0. - Michael Somos, Jul 04 2014
From Peter Bala, Jul 29 2014: (Start)
a(n) = F(n), where the function F(x) := Integral_{0..infinity} e^(-u)*(1 + u)^x du smoothly interpolates this sequence to all real values of x. Note that F(-1) = G and for n = 2,3,... we have F(-n) = (-1)^n/(n-1)! *( A058006(n-2) - G ), where G = 0.5963473623... denotes Gompertz's constant - see A073003.
a(n) = n!*e - e*( Sum_{k >= 0} (-1)^k/((n + k + 1)*k!) ).
(End)
a(n) = hypergeometric_U(1, n+2, 1). - Peter Luschny, Nov 26 2014
a(n) ~ exp(1-n)*n^(n-1/2)*sqrt(2*Pi). - Vladimir Reshetnikov, Oct 27 2015
a(n) = A038155(n+2)/A000217(n+1). - Anton Zakharov, Sep 08 2016
a(n) = round(exp(1)*n!), n > 1 - Simon Plouffe, Jul 28 2020
a(n) = KummerU(-n, -n, 1). - Peter Luschny, May 10 2022
a(n) = (e/(2*Pi))*Integral_{x=-oo..oo} (n+1+i*x)!/(1+i*x) dx. - David Ulgenes, Apr 18 2023
Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * a(i) = n!. - Werner Schulte, Apr 03 2024

Additional comments from Michael Somos

cp's OEIS Frontend

A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

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A000522 Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

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A051396 a(n) = (2*n-2)*(2*n-3)*a(n-1)+1.

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A009628 Expansion of e.g.f.: sinh(x)/(1+x).

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A352659 a(n) = n! * Sum_{k=0..floor(n/3)} 1 / (3*k)!.

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A051396 a(n) = (2n-2)(2n-3)a(n-1)+1.

A080227 a(n) = na(n-1) + (1/2)(1+(-1)^n), a(0)=0.