A001563 -id:A001563 - cp's OEIS frontend

A182386 a(0) = 1, a(n) = 1 - n * a(n-1).

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

Offset: 0

Michael Somos, Apr 27 2012

sign

Hankel transform is A055209.

			G.f. = 1 + x^2 - 2*x^3 + 9*x^4 - 44*x^5 + 265*x^6 - 1854*x^7 + 14833*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450
Shirali Kadyrov, Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.

Cf. A000166, A001563, A055209.

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/(1+x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 11 2018

a:= proc(n) option remember; `if`(n=0, 1, 1-n*a(n-1)) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 19 2015

a[n_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x]
Table[a[n] /. x -> 1, {n, 0, 20}] (* Gerry Martens , May 05 2016 *)
a[0] = 1; a[n_] := a[n] = 1 - n a[n - 1]; Table[a@ n, {n, 0, 22}] (* Michael De Vlieger, May 05 2016 *)

{a(n) = if( n<1, n==0, 1 - n * a(n-1))};

A182386 = lambda n: hypergeometric([-n, 1], [], 1)
print([simplify(A182386(n)) for n in range(23)]) # Peter Luschny, Oct 19 2014

a(n+2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A001563(k+1) for k = 0, 1, ..., n.
E.g.f.: exp(x) / (1 + x).
a(n) = (-1)^n * A000166(n).
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 13 2012
E.g.f.: exp(x) / (1 + x) = 1/(1 - x^2/2 + x^3/(U(0) + 2*x))  where U(k)= k^2 + k*(4-x) - 2*x + 3 + x*(k+1)*(k+3)^2/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 16 2012
G.f.: 1/Q(0) where Q(k) =1 + 4*k*x - x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 10 2013
G.f.: 1/Q(0), where Q(k)= 1 + (k+1)^2*(x) - x/(1 - x*(k+1)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
a(n) = (-1)^n*Gamma(n+1,-1)*exp(-1), where Gamma(a,x) is the incomplete gamma function. - Ilya Gutkovskiy, May 05 2016
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)) for all n>=0. - Michael Somos, Jun 26 2018

A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

Original entry on oeis.org

5, 6, 14, 48, 216, 1200, 7920, 60480, 524160, 5080320, 54432000, 638668800, 8143027200, 112086374400, 1656387532800, 26153487360000, 439378587648000, 7825123418112000, 147254595231744000, 2919482409811968000, 60822550204416000000, 1328364496464445440000

Offset: 0

Bruno Berselli, Feb 22 2017

C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), page 240 (Example 9.57 gives the recurrence).

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A229039.

Cf. sequences with formula (n + k)*n!: A052521 (k=-5), A282822 (k=-4), A052520 (k=-3), A052571 (k=-2), A062119 (k=-1), A001563 (k=0), A000142 (k=1), A001048 (k=2), A052572 (k=3), A052644 (k=4), this sequence (k=5).

A282466:= func< n | (n+5)*Factorial(n) >; // G. C. Greubel, May 14 2025

RecurrenceTable[{a[0] == 5, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}] (* or *)
Table[(n + 5) n!, {n, 0, 30}]

def A282466(n): return (n+5)*factorial(n) # G. C. Greubel, May 14 2025

E.g.f.: (5 - 4*x)/(1 - x)^2.
a(n) = (n + 5)*n!.
a(n) = 2*A229039(n) for n>0.
From Amiram Eldar, Nov 06 2020: (Start)
Sum_{n>=0} 1/a(n) = 9*e - 24.
Sum_{n>=0} (-1)^n/a(n) = 24 - 65/e. (End)

A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

Original entry on oeis.org

1, 1, -1, 2, -4, 2, 6, -18, 18, -6, 24, -96, 144, -96, 24, 120, -600, 1200, -1200, 600, -120, 720, -4320, 10800, -14400, 10800, -4320, 720, 5040, -35280, 105840, -176400, 176400, -105840, 35280, -5040, 40320, -322560, 1128960, -2257920, 2822400, -2257920, 1128960, -322560, 40320, 362880, -3265920

Offset: 0

N. J. A. Sloane

Triangle T(n,k), read by rows: given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 14 2005

			Triangle begins:
   1;
   1,  -1;
   2,  -4,   2;
   6, -18,  18,  -6;
  24, -96, 144, -96, 24;
  ...
x^3 = 6*LaguerreL(0,x) - 18*LaguerreL(1,x) + 18*LaguerreL(2,x) - 6*LaguerreL(3,x).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials

Columns include (essentially) A000142, A001563, A001804, A001805, A001806, A001807.
Cf. A000165 (row sum of absolute values).
Cf. A136572.

[[(-1)^k*Factorial(n)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

row[n_] := Table[ a[n, k], {k, 0, n}] /. SolveAlways[ x^n == Sum[ a[n, k]*LaguerreL[k, x], {k, 0, n}], x] // First; (* or, after Vladeta Jovovic: *) row[n_] := Table[(-1)^k*n!*Binomial[n, k], {k, 0, n}]; Table[ row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Oct 05 2012 *)

for(n=0,10, for(k=0,n, print1((-1)^k*n!*binomial(n,k), ", "))) \\ G. C. Greubel, Feb 06 2018

T(n, k) = (-1)^k*n!*binomial(n, k). - Vladeta Jovovic, May 11 2003
Sum_{k>=0} T(n, k)*T(m, k) = (n+m)!. - Philippe Deléham, Feb 14 2005
Unsigned sequence = A136572 * A007318 - Gary W. Adamson, Jan 07 2008
A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. - Gerald McGarvey, Aug 20 2009

More terms from Vladeta Jovovic, May 11 2003

A052582 a(n) = 2nn!.

Original entry on oeis.org

0, 2, 8, 36, 192, 1200, 8640, 70560, 645120, 6531840, 72576000, 878169600, 11496038400, 161902540800, 2440992153600, 39230231040000, 669529276416000, 12093372555264000, 230485453406208000, 4622513815535616000, 97316080327065600000, 2145819571211796480000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Total number of pairs (a_i,a_(i+1)) in all permutations on [n] such that a_i,a_(i+1) are consecutive integers. - David Callan, Nov 04 2003
Number of permutations of {1,2,...,n+2} such that there is exactly one entry between the entries 1 and 2. Example: a(2)=8 because we have 1324, 1423, 2314, 2413, 3142, 4132, 3241 and 4231. - Emeric Deutsch, Apr 06 2008
Number of permutations of 0 to n distinct letters (ABC...) 1 times ("-" (0), A (1), AB (1-1), ABC (1-1-1), ABCD (1-1-1-1 )etc...) and one after the other to resemble motif:( "-",... BB (0-2), ABB (1-2-0), AABB (2-2-0-0), AAABB (3-2-0-0-0) AAAABB (4-2-0-0-0-0), AAAAABB (5-2-0-0-0-0-0), AAAAAABB (6-2-0-0-0-0-0-0), etc... 0 fixed point (or free fixed point). Example: if ABC (1-1-1) and motif ABB (1-2-0) then 2 * 0 (free) fixed point, if ABCD (1-1-1-1), and motif AABB (2-2-0-0) then 8 * 0 (free) fixed point, if ABCDE (1-1-1-1-1), and motif AAABB (3-2-0-0-0), then 36 * 0 (free) fixed point, if ABCDEF (1-1-1-1-1-1), and motif AAAABB (4-2-0-0-0-0), then 192 * 0 (free) fixed point, if ABCDEFG (1-1-1-1-1-1-1), and motif AAAAABB (5-2-0-0-0-0-0), then 1200 * 0 (free) fixed point, etc... - Zerinvary Lajos, Dec 07 2009

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 526.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A001339, A001563, A001620, A007808, A138770, A000142, A000290, A091725, A099285.

a052582 n = a052582_list !! n
a052582_list =  0 : 2 : zipWith
   div (zipWith (*) (tail a052582_list) (drop 2 a000290_list)) [1..]
-- Reinhard Zumkeller, Nov 12 2011

spec := [S,{S=Prod(Sequence(Z),Sequence(Z),Union(Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ 2 x / (1 - x)^2, {x, 0, n}]]; (* Michael Somos, Oct 20 2011 *)
a[ n_] := If[ n<0, 0, 2 n n!]; (* Michael Somos, Oct 20 2011 *)

{a(n) = if( n<0, 0, 2 * n * n!)}; /* Michael Somos, Oct 20 2011 */

E.g.f.: 2*x / (1 - x)^2.
Recurrence: {a(0)=0, a(1)=2, (-n^2-2*n-1)*a(n)+a(n+1)*n=0.}.
a(n) = A138770(n+2,1). - Emeric Deutsch, Apr 06 2008
a(n) = A001339(n) - A007808(n). - Michael Somos, Oct 20 2011
a(n) = (a(n-1)^2 - 2 * a(n-2)^2 + a(n-2) * a(n-3) - 4 * a(n-1) * a(n-3)) / (a(n-2) - a(n-3)) if n>2. - Michael Somos, Oct 20 2011
a(n) = 2*n*n!. - Gary Detlefs, Sep 16 2010
a(n+1) = a(n) * (n+1)^2 / n. - Reinhard Zumkeller, Nov 12 2011
0 = a(n)*(+a(n+1) -4*a(n+2) +a(n+3)) +a(n+1)*(+2*a(n+1) -a(n+3)) + a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, Jun 26 2017
From Amiram Eldar, Feb 14 2021: (Start)
Sum_{n>=1} 1/a(n) = (Ei(1) - gamma)/2 = (A091725 - A001620)/2, where Ei(x) is the exponential integral.
Sum_{n>=1} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/2 = (A001620 + A099285)/2. (End)
a(n) = 2 * A001563(n). - Alois P. Heinz, Sep 03 2024

A083410 a(n) = A083385(n)/n.

Original entry on oeis.org

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A094258 a(1) = 1, a(n+1) = n*n! for n >= 1.

Original entry on oeis.org

1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000, 24728016011107368960000, 594596384994354462720000

Offset: 1

Amarnath Murthy, Apr 26 2004

nonn

The old definition was: "a(1) = 1, a(n+1) = n*(a(1) + a(2) + ... + a(n))."
a(n) is the number of positive integers k <= n! such that k is not divisible by n. It is also the number of rationals in (0,1] which can be written in the form m/n! but not in the form m/(n-1)!. - Jonathan Sondow, Aug 14 2006
Also, the number of monomials in the determinant of an n X n symbolic matrix with only one zero entry. The position of the zero in the matrix is not important. - Artur Jasinski, Apr 02 2008
From Zak Seidov, Jun 21 2005: (Start)
The number of integers that use each of the decimal digits 0 through n exactly once is the finite sequence 1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, because there are (n+1)! permutations of decimal digits 0 through n, from which we remove the n! permutations with leading zero and get n*n! = total number of integers that use each of the decimal digits from 0 through n exactly once. For n=0 we have 1 integer (0) which uses zero exactly once, hence a(0)=1 by definition.
This sequence is finite because there are only 10 decimal digits. With the initial 1 replaced by 0, we get the initial terms of A001563, which is infinite. Cf. A109075 = number of primes which use each of the decimal digits from 0 through n exactly once. (End)
Partial sums yield factorial numbers A000142(n) = n! = (1, 2, 6, 24, 120, 720, ...). - Vladimir Joseph Stephan Orlovsky, Jun 27 2009

			a(1) = 1;
a(2) = 1*a(1) = 1;
...
a(7) = 6*(a(1) + a(2) + ... + a(6)) = 6*(1 + 1 + 4 + 18 + 96 + 600) = 4320.

Harvey P. Dale, Table of n, a(n) for n = 1..400
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.

Up to the offset and initial value, the same as A001563, cf. formula.

Cf. A109075.

A094258 := proc(n) option remember: if n = 1 then 1; else (n-1)*add(A094258(i),i=1..n-1) ; fi ; end: seq(A094258(n),n=1..24) ; # R. J. Mathar, Jul 27 2007

a=s=1;lst={a};Do[a=s*n-s;s+=a;AppendTo[lst,a],{n,2,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 27 2009 *)
Module[{lst={1}},Do[AppendTo[lst,n*Total[lst]],{n,30}];lst] (* Harvey P. Dale, Jul 01 2012 *)

A094258(n)=(n-1)!*(n-(n>1)) \\ M. F. Hasler, Oct 21 2012

a(n+1) = n*n! = A001563(n) for n>=1.
From Jonathan Sondow, Aug 14 2006: (Start)
a(n) = n! - (n-1)! for n >= 2.
a(n) = n! - a(n-1) - a(n-2) - ... - a(1). with a(1) = 1. (End)
a(n) = A094304(n+1) = A001563(n-1) for n >= 2. - Jaroslav Krizek, Oct 16 2009
G.f.: 1/Q(0), where Q(k)= 1 + x/(1-x) - x/(1-x)*(k+2)/(1 - x/(1-x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: W(0)*(1-sqrt(x)), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 18 2013

Edited by Mark Hudson, Jan 05 2005
More terms from R. J. Mathar, Jul 27 2007
Edited by M. F. Hasler, Oct 21 2012
Edited by Jon E. Schoenfield, Jan 17 2015
Definition simplified by M. F. Hasler, Jun 28 2016

A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 24, 12, 8, 6, 120, 60, 40, 30, 24, 720, 360, 240, 180, 144, 120, 5040, 2520, 1680, 1260, 1008, 840, 720, 40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040, 362880, 181440, 120960, 90720, 72576, 60480, 51840, 45360, 40320, 3628800, 1814400, 1209600, 907200, 725760, 604800, 518400, 453600, 403200, 362880

Offset: 1

Amarnath Murthy, Apr 29 2004

The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.
T(n,k) is the number of length k cycles in all permutations of {1..n}.
Second diagonal gives A001048(n). - Anton Zakharov, Oct 24 2016
T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - Dennis P. Walsh, May 24 2020

			Triangle begins as:
      1;
      2,     1;
      6,     3,     2;
     24,    12,     8,     6;
    120,    60,    40,    30,   24;
    720,   360,   240,   180,  144,  120;
   5040,  2520,  1680,  1260, 1008,  840,  720;
  40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
  ...
T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000142, A000254, A001563, A001710, A002301.
Cf. A061579, A094307, A110468, A133799.
Cf. A129825. - Johannes W. Meijer, Jun 18 2009

Maple
```
seq(seq(n!/k, k=1..n), n=1..10);
```

Table[n!/k, {n,10}, {k,n}]//Flatten
Table[n!/Range[n], {n,10}]//Flatten (* Harvey P. Dale, Mar 12 2016 *)

E.g.f. for column k: x^k/(k*(1-x)).

T(n,k)*k = n*n! = A001563(n).

More terms from Philippe Deléham, Jun 11 2005

A002468 The game of Mousetrap with n cards: the number of permutations of n cards having at least one hit after 2.

Original entry on oeis.org

0, 0, 1, 3, 13, 65, 397, 2819, 22831, 207605, 2094121, 23205383, 280224451, 3662810249, 51523391965, 776082247979, 12463259986087, 212573743211549, 3837628837381201, 73108996989052175, 1465703611456618891, 30847249002794047793, 679998362512214208901, 15668677914172813691699, 376683592679293811722735

Offset: 1

N. J. A. Sloane

The subsequence of primes begins: 3, 13, 397, 2819, no more through a(19). - Jonathan Vos Post, Feb 01 2011

R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
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).

Joerg Arndt, Table of n, a(n) for n = 1..102
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002467, A002468, A002469, A028306, etc.

a[n_] := (n-2)*(n-2)!-(n-4)*Subfactorial[n-3]-(n-3)*Subfactorial[n-2]; a[1]=a[2]=0; a[3]=1; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Dec 12 2014 *)

a(n) = A001563(n) - A002469(n+2). (corrected by Sean A. Irvine and Joerg Arndt, Feb 10 2014)

Added two more terms, Joerg Arndt, Feb 15 2014

A094793 a(n) = (1/n!)*A001688(n).

Original entry on oeis.org

9, 53, 181, 465, 1001, 1909, 3333, 5441, 8425, 12501, 17909, 24913, 33801, 44885, 58501, 75009, 94793, 118261, 145845, 178001, 215209, 257973, 306821, 362305, 425001, 495509, 574453, 662481, 760265, 868501, 987909, 1119233, 1263241

Offset: 0

Benoit Cloitre, Jun 11 2004

Number of injections from {1,2,3,4} to {1,2,...,n} with no fixed points. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006

In general (cf. A094792, A094794, A094795, etc.), the number of injections [k] -> [n] with no fixed points is given by Sum_{i=0..k} (-1)^i*binomial(k,i)*(n-i)!/(n-k)!, which is equal to (1/n!)*f_k(n) where f_k(n) gives the k-th differences of factorial numbers. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001563, A001564, A001565, A001688, A001689, A023043.

Cf. A094791, A094792, A094794, A094795.

LinearRecurrence[{5,-10,10,-5,1},{9,53,181,465,1001},40] (* Harvey P. Dale, May 23 2016 *)

a(n) = n^4 + 6*n^3 + 17*n^2 + 20*n + 9.
a(n) = Sum_{i=0..4} (-1)^i*binomial(4,i)*(n-i)!/(n-4)!. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006
G.f.: -(x^4+6*x^2+8*x+9) / (x-1)^5. - Colin Barker, Jun 16 2013
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Fung Lam, Apr 17 2014
P-recursive: n*a(n) = (n+5)*a(n-1) - a(n-2) with a(0) = 9 and a(1) = 53. Cf. A094791. - Peter Bala, Jul 25 2021

A094794 a(n) = (1/n!)*A001689(n).

Original entry on oeis.org

44, 309, 1214, 3539, 8544, 18089, 34754, 61959, 104084, 166589, 256134, 380699, 549704, 774129, 1066634, 1441679, 1915644, 2506949, 3236174, 4126179, 5202224, 6492089, 8026194, 9837719, 11962724, 14440269, 17312534, 20624939, 24426264

Offset: 0

Benoit Cloitre, Jun 11 2004

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001563, A001564, A001565, A001688, A001689, A023043, A096307.

Cf. A094791, A094792, A094793, A094795.

Table[n^5+10n^4+45n^3+100n^2+109n+44,{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{44,309,1214,3539,8544,18089},30]

a(n)=n^5+10*n^4+45*n^3+100*n^2+109*n+44 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), with a(0)=44, a(1)=309, a(2)=1214, a(3)=3539, a(4)=8544, a(5)=18089. - Harvey P. Dale, Jul 25 2012
G.f.: (x^5 + 10*x^3 + 20*x^2 + 45*x + 44) / (x-1)^6. - Colin Barker, Jun 15 2013
P-recursive: n*a(n) = (n+6)*a(n-1) - a(n-2) with a(0) = 44 and a(1) = 309. Cf. A094791 and A096307. - Peter Bala, Jul 25 2021

cp's OEIS Frontend

A182386 a(0) = 1, a(n) = 1 - n * a(n-1).

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A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

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A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

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A052582 a(n) = 2*n*n!.

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A083410 a(n) = A083385(n)/n.

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A094258 a(1) = 1, a(n+1) = n*n! for n >= 1.

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A052582 a(n) = 2nn!.