A082425 -id:A082425 - 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

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.

A193657 First difference of A002627.

Original entry on oeis.org

1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A193649, A094727, A082425, A114870, A176305, A296964.

a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014

q[n_, k_] := n + k + 1;  (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 20 2012 *)

a(n) = { sum(k=0, n, if (k <= n-2, binomial(n,k)*(k+1)!, binomial(n,k)^2*k!));} \\ Michel Marcus, Feb 07 2013

def A193657():
    a = 2; b = 7; c = 31; n = 3
    yield 1
    while True:
        yield a
        n += 1
        a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014

E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
a(n) = A296964(n+1) - A296964(n) for n >= 2.
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
Second column of A176305. (End)

Simpler definition by Peter Luschny, May 30 2014

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

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

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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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A383436 a(1) = 1; a(n) = 2 + 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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A193657 First difference of A002627.

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

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