A007808 -id:A007808 - cp's OEIS frontend

A056542 a(n) = n*a(n-1) + 1, a(1) = 0.

0, 1, 4, 17, 86, 517, 3620, 28961, 260650, 2606501, 28671512, 344058145, 4472755886, 62618582405, 939278736076, 15028459777217, 255483816212690, 4598708691828421, 87375465144740000, 1747509302894800001, 36697695360790800022, 807349297937397600485

Offset: 1

Henry Bottomley, Jun 20 2000

For n >= 2 also operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of loop repetitions of the j search loop in step L2. - Hugo Pfoertner, Feb 06 2003
More directly: sum over all permutations of length n-1 of the product of the length of the first increasing run by the value of the first position. The recurrence follows from this definition. - Olivier Gérard, Jul 07 2011
This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - T. D. Noe, Jul 07 2005
This sequence also represents the number of subdeterminant evaluations when calculation a determinant by Laplace recursive method. - Reinhard Muehlfeld, Sep 14 2010
Also, a(n) equals the number of non-isomorphic directed graphs of n+1 vertices with 1 component, where each vertex has exactly one outgoing edge, excluding loops and cycle graphs. - Stephen Dunn, Nov 30 2019

			a(4) = 4*a(3) + 1 = 4*4 + 1 = 17.
Permutations of order 3 .. Length of first run * First position
123..3*1
132..2*1
213..1*2
231..2*2
312..1*3
321..1*3
a(4) = 3+2+2+4+3+3 = 17. - _Olivier Gérard_, Jul 07 2011

D. E. Knuth: The Art of Computer Programming, Volume 4, Combinatorial Algorithms, Volume 4A, Enumeration and Backtracking. Pre-fascicle 2B, A draft of section 7.2.1.2: Generating all permutations. Available online; see link.

T. D. Noe, Table of n, a(n) for n = 1..100
D. E. Knuth, TAOCP Vol. 4, Pre-fascicle 2b (generating all permutations).
Tom Muller, Prime and Composite Terms in Sloane's Sequence A056542, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3. [Includes factorizations of a(1) through a(50)]
Hugo Pfoertner, FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation.
R. Sedgewick, Permutation generation methods, Computing Surveys, 9 (1977), 137-164.
Sam Wagstaff, Factorizations of a(51) through a(90)

Cf. A079751 (same recursion formula, but starting from a(3)=0), A038155, A038156, A080047, A080048, A080049.
Cf. A007808, A002627.
Equals the row sums of A162995 triangle (n>=2). - Johannes W. Meijer, Jul 21 2009
Cf. A073333, A002627, A185108.
Cf. A329426, A329427.
Cf. A070213 (indices of primes).

a056542 n = a056542_list !! (n-1)
a056542_list = 0 : map (+ 1) (zipWith (*) [2..] a056542_list)
-- Reinhard Zumkeller, Mar 24 2013

[n le 2 select n-1 else n*Self(n-1)+1: n in [1..20]]; // Bruno Berselli, Dec 13 2013

tmp=0; Join[{tmp}, Table[tmp=n*tmp+1, {n, 2, 100}]] (* T. D. Noe, Jul 12 2005 *)
FoldList[ #1*#2 + 1 &, 0, Range[2, 21]] (* Robert G. Wilson v, Oct 11 2005 *)

a(n) = floor((e-2)*n!).
a(n) = A002627(n) - n!.
a(n) = A000522(n) - 2*n!.
a(n) = n! - A056543(n).
a(n) = (n-1)*(a(n-1) + a(n-2)) + 2, n > 2. - Gary Detlefs, Jun 22 2010
1/(e - 2) = 2! - 2!/(1*4) - 3!/(4*17) - 4!/(17*86) - 5!/(86*517) - ... (see A002627 and A185108). - Peter Bala, Oct 09 2013
E.g.f.: (exp(x) - 1 - x) / (1 - x). - Ilya Gutkovskiy, Jun 26 2022

More terms from James Sellers, Jul 04 2000

A074143 a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, 18144000, 219542400, 2874009600, 40475635200, 610248038400, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 24329020081766400000, 536454892802949120000

Offset: 1

Amarnath Murthy, Aug 28 2002

nonn

a(n) is also the number of elements of the alternating semigroup (A^c_n) for F(n, p) if p = n - 1 (cf. A001710). - Bakare Gatta Naimat, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets
Stephen Lipscomb, Symmetric inverse semigroups, Mathematical surveys and monographs, Vol. 46 Amer. Math. Soc. (1996).
Michael Penn, Australian Mathematical Olympiad 2018 Question 5, Youtube video, 2020.

A diagonal of A254040.
Cf. A001563, A001710.
Cf. A007808, A082425, A082427, A082428, A082430, A383436, A383437.
Cf. A001620, A091725, A099285.

[Numerator(Factorial(n)/2*n): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014

seq(sum(mul(j,j=3..n), k=1..n), n=1..19); # Zerinvary Lajos, Jun 01 2007
a := n -> `if`(n=1,1,n!*n/2): seq(a(n), n=1..19); # Peter Luschny, Jan 22 2016

A074143[1] = 1; A074143[n_] := A074143[n] = n * Sum[a[k], {k, n - 1}]; Array[A074143, 20] (* T. D. Noe, Apr 05 2011 *)
Table[Numerator[n!/2 n], {n, 21}] (* Vincenzo Librandi, Apr 15 2014 *)

def b(n): return 1/2 if (n==1) else n^2*b(n-1)/(n-1)
def A074143(n): return b(n) + int(n==1)/2
[A074143(n) for n in range(1,41)] # G. C. Greubel, Nov 29 2022

a(n) = n^2 * a(n-1)/(n-1) for n > 2.
a(n) = n*ceiling(n!/2) = n*A001710(n) = ceiling(A001563(n)/2). - Henry Bottomley, Nov 27 2002
a(n) = ((n+1)!-n!)/2 for n > 1. - Vladimir Joseph Stephan Orlovsky, Apr 03 2011
G.f.: (U(0) + x)/(2*x) where U(k) = 1 - 1/(k+1 - x*(k+1)^2*(k+2)/(x*(k+1)*(k+2) - 1/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2012
G.f.: 1/2 + Q(0), where Q(k)= 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*binomial(n, j)*(j)^(n+1) / (n+1), n > 1, a(1) = 1. - Vladimir Kruchinin, Jun 01 2013
a(n) = numerator(n!/2*n). - Vincenzo Librandi, Apr 15 2014
a(n) is F(n;p) = n^2(n-1)!/2 if p = n-1 in A^c_n. For instance for n=4 and p=n-1: F(4; 4-1)= 4^2(4-1)!/2 = 16*6/2 = 48. - Bakare Gatta Naimat, Nov 18 2015
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: x/2 * (1 + 1/(1-x)^2).
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)
From Amiram Eldar, May 04 2025: (Start)
Sum_{n>=1} 1/a(n) = 2*ExpIntegralEi(1) - 2*gamma - 1 = 2*A091725 - 2*A001620 - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*gamma - 1 - 2*ExpIntegralEi(-1) = 2*A001620 - 1 + 2*A099285. (End)

More terms from Henry Bottomley, Nov 27 2002

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

A082425 a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).

Original entry on oeis.org

1, 1, 5, 27, 169, 1217, 9939, 90871, 920069, 10222989, 123698167, 1619321459, 22805443881, 343835923129, 5525934478859, 94309281772527, 1703461402016269, 32465970250192421, 651123070017747999, 13707854105636799979, 302258183029291439537, 6966331456484621749329

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A001339, A056543, A083746.

Cf. A007808, A074143, A082427, A082428, A082430, A383436, A383437.

[n le 2 select 1 else (n^2*Self(n-1) +1)/(n-1): n in [1..30]]; // G. C. Greubel, Feb 03 2024

a:= n -> n*n!*add(1/(k*(k-1)*k!), k = 2..n): seq(a(n), n = 2..20); # Peter Bala, Jul 09 2008

a[n_]:= a[n]= If[n<3, 1, -1 +n*Sum[a[j], {j,n-1}]];
Table[a[n], {n,40}] (* G. C. Greubel, Feb 03 2024 *)

@CachedFunction # a = A082425
def a(n): return 1 if (n==1) else -1 + n*sum(a(j) for j in range(1,n))
[a(n) for n in range(1,41)] # G. C. Greubel, Feb 03 2024

For n >= 2, a(n) = floor(n*(3-e)*n!).
a(n) = n*A056543(n) - 1, n > 1. - Vladeta Jovovic, Apr 26 2003
From Peter Bala, Jul 09 2008: (Start)
In the following remarks we use an offset of 1, i.e., a(1) = 1, a(2) = 1, a(3) = 5, ... .
  For n >= 2, a(n) = n*n!*Sum_{k = 2..n} 1/(k*(k-1)*k!).
  For n >= 2, a(n) = 3*n*n! - Sum_{k = 0..n} (k+1)!*binomial(n,k).
  Limit_{n -> oo} a(n)/(n*n!) = 3 - e.
  E.g.f.: 1 + t + (3*t - exp(t))/(1-t)^2.
  a(n) = A083746(n+2) - A001339(n).
Recurrence relation: a(1) = 1, a(2) = 1, a(3) = 5, a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n >= 4.
Recurrence relation: a(1) = 1, a(2) = 1, a(n) = (n^2*a(n-1) + 1)/(n-1) for n >= 3.
The recurrence relation x(n) = (n^2*x(n-1) - 1)/(n-1), for n >= 2, has the general solution x(n) = n*n!*x(1) - a(n); particular solutions are A007808 (x(1) = 1) and A001339 (x(1) = 3). (End)

Offset corrected by G. C. Greubel, Feb 03 2024

A082430 a(1)=1; for n > 1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + 4.

Original entry on oeis.org

1, 6, 25, 132, 824, 5932, 48444, 442916, 4484524, 49828044, 602919332, 7892762164, 111156400476, 1675896499484, 26934050884564, 459674468429892, 8302870086014924, 158242935756990316, 3173649989348528004, 66813683986284800084, 1473241731897579841852

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

More generally, if m is an integer and a(1)=1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + m then a(n) has a closed form formula as a(n) = floor/ceiling(n*r(m)*n!) where r(m) = frac(e*m) + 0 or + 1/2 or -1/2 + integer. (See Example section.)

			r(10) = frac(10*e) + 1/2 + 2;
r(12) = frac(12*e) - 1/2 + 3;
r(15) = frac(15*e) + 3;
r(18) = frac(18*e) - 1/2 + 4.

Harvey P. Dale, Table of n, a(n) for n = 1..448

Cf. A007808, A074143, A082425, A082427, A082428, A383436, A383437.

Cf. A001339.

nxt[{n_,t_,a_}]:=Module[{c=t(n+1)+4},{n+1,t+c,c}]; NestList[nxt,{1,1,1},20][[;;,3]] (* Harvey P. Dale, Mar 28 2024 *)

For n >= 2, a(n) = ceiling(n*(19/2 - 4*e)*n!).
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: -4 - 3*x/2 + (-19*x/2 + 4*exp(x))/(1-x)^2.
a(n) = -19*n/2 * n! + 4 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 4)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)

A082427 a(1)=1, a(n) = n * (Sum_{k=1..n-1} a(k)) - 2.

Original entry on oeis.org

1, 0, 1, 6, 38, 274, 2238, 20462, 207178, 2301978, 27853934, 364633318, 5135252562, 77423807858, 1244311197718, 21236244441054, 383579665216538, 7310577148832842, 146617686151591998, 3086688129507199958, 68061473255633759074, 1568654907415559018658

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

Cf. A007808, A074143, A082425, A082428, A082430, A383436, A383437.

Cf. A001339.

Join[{1},Table[Floor[n(11/2-2E)n!],{n,2,20}]] (* Harvey P. Dale, May 09 2013 *)

a(n) = floor(n*(11/2 - 2*e)*n!) for n >= 2.
a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n>3. - Gary Detlefs, Jun 30 2024
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: 2 + 3*x/2 + (11*x/2 - 2*exp(x))/(1-x)^2.
a(n) = 11*n/2 * n! - 2 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) + 2)/(n-1) for n > 2. (End)

Offset changed to 1 by Georg Fischer, May 15 2024

A082428 a(1) = 1; a(n) = 3 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 5, 21, 111, 693, 4989, 40743, 372507, 3771633, 41907033, 507075099, 6638074023, 93486209157, 1409484384213, 22652427603423, 386601431098419, 6982988349215193, 133087542655630737, 2669144605482372003, 56192518010155200063, 1239045022123922161389, 28557037652760872672013

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

Cf. A007808, A074143, A082425, A082427, A082430, A383436, A383437.

Cf. A001339.

for n>=2 a(n) = ceiling(n*(3e-7)*n!).
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: -3 - x + (-7*x + 3*exp(x))/(1-x)^2.
a(n) = -7 * n * n! + 3 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 3)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)

A383436 a(1) = 1; a(n) = 2 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 4, 17, 90, 562, 4046, 33042, 302098, 3058742, 33986022, 411230866, 5383385882, 75816017838, 1143072268942, 18370804322282, 313528393766946, 5663106612415462, 107932149554271158, 2164639221616216002, 45571352034025600042, 1004848312350264480926, 23159361103691809941342

Offset: 1

Seiichi Manyama, Apr 27 2025

nonn

Cf. A007808, A074143, A082425, A082427, A082428, A082430, A383437.

Cf. A001339.

my(N=30, x='x+O('x^N)); Vec(serlaplace(-2-x/2+(-9*x/2+2*exp(x))/(1-x)^2))

E.g.f.: -2 - x/2 + (-9*x/2 + 2*exp(x))/(1-x)^2.
a(n) = -9*n/2 * n! + 2 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 2)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3.

A383437 a(1) = 1; a(n) = 5 + n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 7, 29, 153, 955, 6875, 56145, 513325, 5197415, 57749055, 698763565, 9147450305, 128826591795, 1942308614755, 31215674165705, 532747505761365, 9622751822814655, 183398328858349895, 3678155373214684005, 77434849962414400105, 1707438441671237522315

Offset: 1

Seiichi Manyama, Apr 27 2025

nonn

Cf. A007808, A074143, A082425, A082427, A082428, A082430, A383436.

Cf. A001339.

my(N=30, x='x+O('x^N)); Vec(serlaplace(-5-2*x+(-12*x+5*exp(x))/(1-x)^2))

E.g.f.: -5 - 2*x + (-12*x + 5*exp(x))/(1-x)^2.
a(n) = -12 * n * n! + 5 * Sum_{k=0..n} (k+1)! * binomial(n,k) for n > 1.
a(n) = (n^2 * a(n-1) - 5)/(n-1) for n > 2.
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3.

A141827 a(n) = (n^3*a(n-1) - 1)/(n - 1) for n >= 2, with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 31, 418, 8917, 278656, 12037939, 688168846, 50334635593, 4586743668412, 509638185379111, 67832842473959674, 10655922890454756061, 1950921882527424922168, 411794588127327229725307, 99271909637837814308779366, 27107849458438912493917352209

Offset: 0

Peter Bala, Jul 09 2008, Oct 06 2008

For related recurrences of the form a(n) = (n^k*a(n-1)-1)/(n-1) see A001339, A007808 (both k = 2) and A141828 (k = 4). 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). See A000522 for further properties of difference divisibility sequences.
From Peter Bala, Jul 02 2016: (Start)
For k = 1,2,3,... and x in Z, the recurrence equation a(n) = (n^k*a(n-1) - 1)/(n - 1) with starting value a(1) = x produces an integer sequence. This is because the sequence also satisfies the second-order recurrence a(n) = (1 + (n^k - 1)/(n - 1))*a(n-1) - (n - 1)^(k-1)*a(n-2) with integer starting values a(1) = x, a(2) = x*2^k - 1. Here we take k = 3 and x = 4.
The solution to the recurrence is a(n) = n*n!^(k-1)*( x - Sum_{i = 2..n} 1/(i*(i-1)*i!^(k-1)) ). Hence limit (n -> inf) a(n)/(n*n!^(k-1)) equals the constant x - Sum_{i = 2..inf} 1/(i*(i-1)*i!^(k-1)). Note that the sequence b(n) := n*n!^(k-1) satisfies the same second-order recurrence but with starting values b(1) = 1, b(2) = 2^k. From this observation one can get a generalized continued fraction expansion for a(n)/b(n) and hence, by going to the limit, for the constant x - Sum_{i = 2..n} 1/(i*(i-1)*i!^(k-1)). See, for example, A130820.  (End)

Cf. A001339, A007808, A070910, A082425, A096789, A130820, A141828.

a(n) := n -> n!^2*sum((n-k+1)*(k+1)/k!^2, k = 0..n): seq(a(n), n = 0..16);

a[0] = 1; a[1] = 4; a[n_] := a[n] = (n^3 a[n - 1] - 1)/(n - 1); Table[a@ n, {n, 0, 14}] (* or *)
Table[n!^2 Sum[(n - k + 1) (k + 1)/k!^2, {k, 0, n}], {n, 0, 14}] (* or *)
Table[n n!^2 (4 - Sum[ 1/(k!^2*k*(k - 1)), {k, 2, n}]), {n, 0, 14}] /. 0 -> 1 (* Michael De Vlieger, Jul 03 2016 *)

Sum_{n = 0..inf} a(n)*x^n/n!^2 = 1/(1 - x)^2*Sum_{n = 0..inf} (n + 1)*x^n/n!^2.
a(n) = n!^2*Sum_{k = 0..n} (n - k + 1)(k + 1)/k!^2.
a(n) = n*n!^2*(4 - Sum_{k = 2..n} 1/(k!^2*k*(k - 1))).
Congruence property: a(n) == (1 + n + n^2) (mod n^3).
The recurrence a(n) = (n^2 + n + 2)*a(n-1) - (n - 1)^2*a(n-2), n >= 2, shows that a(n) is always a positive integer. The sequence b(n) := n*n!^2 also satisfies the same recurrence with b(0) = 0, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/(n*n!^2) = 4 - 1^2/(8 - 2^2/(14 - 3^2/(22 -...-(n - 1)^2/(n^2 + n + 2)))), for n > 1. a(n)*b(n+1) - b(n)*a(n+1) = n!^2.
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 0..inf} (n + 1)/n!^2 = BesselI(0,2) + BesselI(1,2) = 3.87022 21569 ..., using the values of the modified Bessel function, BesselI(0, 2) = 2.27958 53023 ... and BesselI(1, 2) = 1.59063 68546 ... (see A070910 and A096789; Cf. A130820). This yields the continued fraction expansion BesselI(0,2) + BesselI(1,2) = 4 - 1^2/(8 - 2^2/(14 - 3^2/(22 -...-(n - 1)^2/(n^2 + n + 2 - ... )))).
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 1..inf} (n + n^2)/n!^2 = Sum_{n = 1..inf} n^3/n!^2 = 1/2 * Sum_{n = 1..inf} n^4/n!^2.
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 0..inf} A001405 (n)/n!.
Limit_{n -> infinity} a(n)/(n*n!^2) = 1 + Sum_{n = 0..inf} 1/(Product_{k = 0..n} A008619(k)).

a(15)-a(16) from Jason Yuen, Jan 31 2025

cp's OEIS Frontend

A056542 a(n) = n*a(n-1) + 1, a(1) = 0.

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A074143 a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).

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

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A082425 a(1)=1, a(n) = -1 + n*Sum_{j=1..n-1} a(j).

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A082430 a(1)=1; for n > 1, a(n) = n*(a(n-1) + a(n-2) + ... + a(2) + a(1)) + 4.

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A082427 a(1)=1, a(n) = n * (Sum_{k=1..n-1} a(k)) - 2.

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A082428 a(1) = 1; a(n) = 3 + n * Sum_{k=1..n-1} a(k).

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A052582 a(n) = 2nn!.