A091725 -id:A091725 - cp's OEIS frontend

A061206 a(n) = total number of occurrences of the consecutive pattern 1324 in all permutations of [n+3].

1, 10, 90, 840, 8400, 90720, 1058400, 13305600, 179625600, 2594592000, 39956716800, 653837184000, 11333177856000, 207484333056000, 4001483566080000, 81096733605888000, 1723305589125120000, 38318206628782080000, 889833909490606080000, 21543347282404147200000

Offset: 1

Melvin J. Knight (knightmj(AT)juno.com), May 30 2001

nonn

a(n) is the number of sequences of n+3 balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 27 2017

			a(4)=840 because 4*(7!)/24 = 4*7*6*5 = 840.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

Cf. A000142, A001286, A001339, A001563, A005990, A264173.

Cf. A001620, A091725, A099285.

[n*Factorial(n+3)/24: n in [1..20]]; // Vincenzo Librandi, Oct 11 2011

a := n -> n!*binomial(-n,4): seq(a(n),n=1..20); # Peter Luschny, Apr 29 2016

Array[# (# + 3)!/24 &, 20] (* or *) Array[#!*Binomial[-#, 4] &, 20] (* Michael De Vlieger, Sep 30 2017 *)

a(n) = n*(n+3)!/24; \\ Altug Alkan, Oct 08 2017

[binomial(n,4)*factorial (n-3) for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

a(n) = n*(n+3)!/24.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i) * x^(k-j), then a(n-3) = (-1)^n*f(n,4,-2), (n >= 4). - Milan Janjic, Mar 01 2009
E.g.f.: x/(1-x)^5. (This was initiated by e-mail exchange with Gary Detlefs.) - Wolfdieter Lang, May 28 2010
a(n) = ((n+4)!/6) * Sum_{k=1..n} (k+2)!/(k+4)!. - Gary Detlefs, Aug 05 2010
a(n) = Sum_{k>0} k * A264173(n+3,k). - Alois P. Heinz, Nov 06 2015
a(n) = n!*binomial(-n,4). - Peter Luschny, Apr 29 2016
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 118/3 - 16*e - 4*gamma + 4*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 - 8/e + 4*gamma - 4*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms from Jason Earls, Jun 12 2001
Corrected by Zerinvary Lajos, Jul 07 2009
More precise definition from Alois P. Heinz, Nov 06 2015

A229837 Decimal expansion of Sum_{n>=1} 1/(n*n!).

Original entry on oeis.org

1, 3, 1, 7, 9, 0, 2, 1, 5, 1, 4, 5, 4, 4, 0, 3, 8, 9, 4, 8, 6, 0, 0, 0, 8, 8, 4, 4, 2, 4, 9, 2, 3, 1, 8, 3, 7, 9, 7, 4, 9, 0, 1, 2, 4, 5, 7, 9, 2, 7, 8, 3, 9, 9, 2, 8, 4, 0, 4, 6, 1, 1, 9, 6, 9, 9, 7, 6, 4, 6, 1, 0, 7, 7, 5, 6, 1, 3, 9, 4, 8, 2, 6, 1, 1, 9, 5, 3, 6, 4, 6, 8, 3, 4, 3, 9, 2, 2, 0, 7

Offset: 1

Jean-François Alcover, Oct 01 2013

			1.3179021514544038948600088442492318379749012457927839928404611969976461...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012, page 17.
Wikipedia, Logarithmic integral function.

Cf. A001008, A001113, A001620, A002805, A091725, A061572, A264806.

evalf(Ei(1)-gamma,120); # Vaclav Kotesovec, May 10 2015

RealDigits[ ExpIntegralEi[1] - EulerGamma, 10, 100] // First

-Euler-real(eint1(-1)) \\ Charles R Greathouse IV, Oct 01 2013

Sum_{n >= 1} 1/(n*n!) = Ei(1)-gamma where Ei is the exponential integral and gamma is Euler's constant.
Also pFq(1,1; 2,2; 1) where pFq is the generalized hypergeometric function.
Also li(e)-gamma, e being the Euler constant (A001113) and li the logarithmic integral function. - Stanislav Sykora, May 09 2015
Continued fraction expansion: Ei(1) - gamma = 1/(1 - 1^3/(5 - 2^3/(11 -...-(n-1)^3/(n^2+n-1) -...))). See A061572. - Peter Bala, Feb 01 2017
From Amiram Eldar, Aug 01 2020: (Start)
Equals Sum_{k>=1} H(k)*k/(k+1)!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals Integral_{x=0..1} (exp(x) - 1)/x dx.
Equals -Integral_{x=0..1} exp(x)*log(x) dx.
Equals -Integral_{x=1..e} log(log(x)) dx. (End)
Equals e * Sum_{k>=1} (-1)^(k+1)*H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jun 25 2021

A062199 Second (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

Original entry on oeis.org

1, 12, 126, 1344, 15120, 181440, 2328480, 31933440, 467026560, 7264857600, 119870150400, 2092278988800, 38532804710400, 746943599001600, 15205637551104000, 324386934423552000, 7237883474325504000, 168600109166641152000, 4093235983656787968000

Offset: 0

Wolfdieter Lang, Jun 19 2001

a(n) is the total number of ascending runs of length 5 over all permutations of {1,2,...,n+5}. a(1) = 12 because we have: [1,2,3,4,6,5], [1,2,3,5,6,4], [1,2,4,5,6,3], [1,3,4,5,6,2], [2,1,3,4,5,6], [2,3,4,5,6,1], [3,1,2,4,5,6], [4,1,2,3,5,6], [5,1,2,3,4,6], [6,1,2,3,4,5], and [1,2,3,4,5,6] which has two runs of length 5. - Geoffrey Critzer, Feb 21 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.
Index entries for sequences related to Laguerre polynomials.

Cf. A001720 (first column of A062140), A264781.

Cf. A001620, A091725, A099285.

[Binomial(n, 5)*Factorial(n-4): n in [5..25]]; // Vincenzo Librandi, Feb 23 2014

Table[Sum[n!/5!, {i, 5, n}], {n, 5, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[(1+5x)/(1-x)^7,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 10 2016 *)

x='x+O('x^30); Vec(serlaplace((1+5*x)/(1-x)^7)) \\ G. C. Greubel, Feb 07 2018

[binomial(n,5)*factorial (n-4) for n in range(5, 22)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1+5*x)/(1-x)^7.
a(n) = A062140(n+1, 1) = (n+1)!*binomial(n+5, 5).
If we define f(n,i,x)= Sum(Sum(binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n-1)=(-1)^(n-1)*f(n,1,-6), (n>=1). [Milan Janjic, Mar 01 2009]
a(n) = Sum_{k>0} k * A264781(n+5,k). - Alois P. Heinz, Nov 24 2015
Assuming offset 1: a(n) = -n!*binomial(-n,5). - Peter Luschny, Apr 29 2016
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 1565/12 - 50*e - 5*gamma + 5*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
Sum_{n>=0} (-1)^n/a(n) = -125/12 + 20/e + 5*gamma - 5*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms from Vincenzo Librandi, Feb 23 2014

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

A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

Original entry on oeis.org

1, 20, 300, 4200, 58800, 846720, 12700800, 199584000, 3293136000, 57081024000, 1038874636800, 19833061248000, 396661224960000, 8299373322240000, 181400588328960000, 4135933413900288000, 98228418580131840000, 2426819753156198400000, 62288373664342425600000

Offset: 4

N. J. A. Sloane

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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).

T. D. Noe, Table of n, a(n) for n = 4..100

Column 4 of A008297.
Column m=4 of unsigned triangle A111596.
Cf. A053495.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-1)*Binomial(n, 4)/6: n in [4..30]]; // G. C. Greubel, May 10 2021

Maple
```
A001755 := n-> n!*binomial(n-1,3)/4!;
```

Table[n!Binomial[n-1, 3]/4!, {n, 4, 25}] (* T. D. Noe, Aug 10 2012 *)

[binomial(n,4)*factorial (n-1)/6 for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: ((x/(1-x))^4)/4!.
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} (binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) ) ) then a(n) = (-1)^n*f(n,4,-4), (n>=4). - Milan Janjic, Mar 01 2009
D-finite with recurrence (-n+4)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 12*(Ei(1) - gamma + 2*e) - 80, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 156*(gamma - Ei(-1)) - 96/e - 88, where Ei(-1) = -A099285. (End)

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

A001777 Lah numbers: a(n) = n! * binomial(n-1, 4)/5!.

Original entry on oeis.org

1, 30, 630, 11760, 211680, 3810240, 69854400, 1317254400, 25686460800, 519437318400, 10908183686400, 237996734976000, 5394592659456000, 126980411830272000, 3101950060425216000, 78582734864105472000, 2062796790182768640000, 56059536297908183040000

Offset: 5

N. J. A. Sloane

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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).

T. D. Noe, Table of n, a(n) for n = 5..100

Column 5 of A008297. Cf. A053495.
Column m=5 of unsigned triangle A111596.
Cf. A001113, A001620, A091725, A099285.

Maple
```
A001777 := n-> n!*binomial(n-1,4)/5!;
```

Table[n! Binomial[n - 1, 4]/5!, {n, 5, 20}] (* T. D. Noe, Aug 10 2012 *)

[binomial(n,5)*factorial (n-1)/factorial (4) for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: ((x/(1-x))^5)/5!.
If we define f(n,i,x) = sum(sum(binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n+1)=(-1)^n*f(n,4,-6), (n>=4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=5} 1/a(n) = 20*(Ei(1) - gamma) - 200*e + 1555/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=5} (-1)^(n+1)/a(n) = 1460*(gamma - Ei(-1)) - 880/e - 2515/3, where Ei(-1) = -A099285. (End)

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

A052571 E.g.f. x^3/(1-x)^2.

Original entry on oeis.org

0, 0, 0, 6, 48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200, 4790016000, 68497228800, 1046139494400, 16999766784000, 292919058432000, 5335311421440000, 102437979291648000, 2067966706950144000

Offset: 0

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

For n >= 3, a(n) = number whose factorial base representation (A007623) begins with digit {n-2} followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 0, 0, 100, 2000, 30000, 400000, 5000000, 60000000, 700000000, 8000000000, 90000000000, A00000000000, B000000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 514.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint, arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for sequences related to factorial base representation

Column 5 of A257503 (apart from zero terms. Equally, row 5 of A257505).
Cf. A000142, A007623, A005990, A090672.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A091725, A099285.

[0,0] cat [n*(n+1)*(n+2)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

spec := [S,{S=Prod(Z,Z,Z,Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
[seq (n*(n+1)*(n+2)*n!,n=0..17)]; # Zerinvary Lajos, Nov 25 2006
a:=n->add((n!),j=1..n-2):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008
G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 01 2009

Table[Sum[n!, {i, 3, n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^3/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 27 2025 *)

(define (A052571 n) (if (< n 2) 0 (* (- n 2) (A000142 n)))) ;; Antti Karttunen, May 07 2015

E.g.f.: x^3/(-1+x)^2.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (1-n^2)*a(n)+(-2+n)*a(n+1)=0}.
For n >= 2, a(n) = (n-2)*n!.
a(n+2) = n*(n+1)*(n+2)*n!. - Zerinvary Lajos, Nov 25 2006
a(n) = 3*A090672(n-2) = 6*A005990(n-2). - Zerinvary Lajos, May 11 2007
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=3} 1/a(n) = 9/4 - e - gamma/2 + Ei(1)/2 = 9/4 - A001113 - (1/2)*A001620 + (1/2)*A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = -1/4 + gamma/2 - Ei(-1)/2 = -1/4 + (1/2)*A001620 + (1/2)*A099285. (End)

A062148 Second (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).

Original entry on oeis.org

1, 14, 168, 2016, 25200, 332640, 4656960, 69189120, 1089728640, 18162144000, 319653734400, 5928123801600, 115598414131200, 2365321396838400, 50685458503680000, 1135354270482432000, 26538906072526848000, 646300418472124416000, 16372943934627151872000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(3) = (3+1)! * binomial(3+6,6) = 24 * 84 = 2016. - _Indranil Ghosh_, Feb 24 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.
Index entries for sequences related to Laguerre polynomials.

Cf. A001725 (first column of A062138).
Appears in the third left hand column of A167556 multiplied by 120. - Johannes W. Meijer, Nov 12 2009
Cf. A001620, A091725, A099285.

[Factorial(n+1)*Binomial(n+6,6): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[Sum[n!/6!, {i, 6, n}], {n, 6, 21}] (* Zerinvary Lajos, Jul 12 2009 *)

a(n)=(n+1)!*binomial(n+6,6) \\ Indranil Ghosh, Feb 24 2017

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062148(n): return f(n+1)*C(n+6,6) # Indranil Ghosh, Feb 24 2017

E.g.f.: (1+6*x)/(1-x)^8.
a(n) = A062138(n+1, 1) = (n+1)!*binomial(n+6, 6).
If we define f(n,i,x)= Sum_{k=i..n}(Sum_{j=i..k}(binomial(k,j) *Stirling1(n,k)* Stirling2(j,i)*x^(k-j))) then a(n-1) = (-1)^(n-1) * f(n,1,-7), (n>=1). - Milan Janjic, Mar 01 2009
Assuming offset 1: a(n) = n!*binomial(-n,6). - Peter Luschny, Apr 29 2016
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 5477/10 - 204*e - 6*gamma + 6*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
Sum_{n>=0} (-1)^n/a(n) = 403/10 - 120/e + 6*gamma - 6*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

A090672 a(n) = (n^2-1)*n!/3.

Original entry on oeis.org

0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000

Offset: 1

N. J. A. Sloane, Dec 18 2003

nonn

a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} |pi(i)-i|, i.e., the total displacement of all letters in all permutations on n letters.
a(n) = number of entries between the entries 1 and 2 in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 123, 1(3)2, 213, 2(3)1, 312, 321; the entries between 1 and 2 are surrounded by parentheses. - Emeric Deutsch, Apr 06 2008
a(n) = Sum_{k=0..n-1} k*A138770(n+1,k). - Emeric Deutsch, Apr 06 2008
a(n) is also the number of peaks in all permutations of {1,2,...,n+1}. Example: a(3)=16 because the permutations 1234, 4123, 3124, 4312, 2134, 4213, 3214, and 4321 have no peaks and each of the remaining 16 permutations of {1,2,3,4} has exactly one peak. - Emeric Deutsch, Jul 26 2009
a(n), for n>=2, is the number of (n+2)-node tournaments that have exactly one triad. Proven by Kadane (1966), see link. - Ian R Harris, Sep 26 2022

D. Daly and P. Vojtechovsky, Displacement of permutations, preprint, 2003.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Ian R. Harris and Ryan P. A. McShane, Counting Tournaments with a Specified Number of Circular Triads, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

Twice A005990.
Cf. A138770.
Cf. A001113, A001620, A091725, A099285.

[(n^2-1)*Factorial(n)/3: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

nn=20;Drop[Range[0,nn]!CoefficientList[Series[ x^3/3/(1-x)^2,{x,0,nn}],x],2]  (* Geoffrey Critzer, Mar 04 2013 *)

a(n) = A052571(n+2)/3 = 2*A005990(n). - Zerinvary Lajos, May 11 2007
a(n) = (n+3)! * Sum_{k=1..n} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010
E.g.f.: (x^3 - 3*x^2)/(3*(x-1)^3). - Geoffrey Critzer, Mar 04 2013
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = (3/2)*(Ei(1) - gamma) - 3*e + 27/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=2} (-1)^n/a(n) = (3/2)*(gamma - Ei(-1)) - 3/4, where Ei(-1) = -A099285. (End)

A001810 a(n) = n!n(n-1)*(n-2)/36.

Original entry on oeis.org

0, 0, 0, 1, 16, 200, 2400, 29400, 376320, 5080320, 72576000, 1097712000, 17563392000, 296821324800, 5288816332800, 99165306240000, 1952793722880000, 40311241850880000, 870722823979008000, 19645683716026368000, 462251381553561600000, 11325158848062259200000

Offset: 0

N. J. A. Sloane

nonn

a(n) is the total number of 3-2-1 patterns in all permutations on [n]. This is because there are n! permutations, binomial(n,3) triples in each one and the probability that a given triple of entries in a random permutation form a 3-2-1 pattern (or any other specified pattern of length 3) is 1/6. - David Callan, Oct 26 2006

Old name was "Coefficients of Laguerre polynomials".

			G.f. = x^3 + 16*x^4 + 200*x^5 + 2400*x^6 + 29400*x^7 + 376320*x^8 + ...

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.
Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
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].
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Index entries for sequences related to Laguerre polynomials.

Cf. A053495, A263771.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n)*n*(n-1)*(n-2)/36: n in [0..20]]; // G. C. Greubel, May 16 2018

[seq(n!*n*(n-1)*(n-2)/36,n=0..30)];
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=0..20) ; # Zerinvary Lajos, Feb 07 2008

Table[n! n*(n-1)*(n-2)/36, {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

for(n=0,20, print1(n!*n*(n-1)*(n-2)/36, ", ")) \\ G. C. Greubel, May 16 2018

[factorial(m) * binomial(m, 3) / 6 for m in range(22)]  # Zerinvary Lajos, Jul 05 2008

a(n) = -A021009(n, 3), n >= 0. a(n) = ((n!/3!)^2)/(n-3)!, n >= 3.
E.g.f.: x^3/(3!*(1-x)^4).
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^(n-1) * f(n,3,-4), (n >= 3). - Milan Janjic, Mar 01 2009
a(n) = Sum_{k>0} k * A263771(n,k). - Alois P. Heinz, Oct 27 2015
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 9*(2*e + gamma - Ei(1) - 4), where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = 63*(gamma - Ei(-1)) - 36*(1/e + 1), where Ei(-1) = -A099285. (End)

Edited by N. J. A. Sloane, Apr 12 2014

cp's OEIS Frontend

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