A383436 -id:A383436 - cp's OEIS frontend

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

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

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)

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.

cp's OEIS Frontend

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

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

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