A000680 -id:A000680 - cp's OEIS frontend

A065140 a(n) = 2^n(2n)!.

1, 4, 96, 5760, 645120, 116121600, 30656102400, 11158821273600, 5356234211328000, 3278015337332736000, 2491291656372879360000, 2301953490488540528640000, 2541356653499348743618560000, 3303763649549153366704128000000, 4995290638118319890456641536000000

Offset: 0

Karol A. Penson, Oct 16 2001

From Enrique Navarrete, Aug 29 2025: (Start)
For n > 0, 1/2*a(n) is the number of ways to seat 2*n people on linearly ordered benches placing an even number of people (>=2) on each bench.
For example, 1/2*a(4)=322560 since the number of ways are (number of people in parentheses):
  1 bench (8): 40320 ways;
  2 benches (6,2): 80640 ways;
  2 benches (4,4): 40320 ways;
  3 benches (4,2,2): 120960 ways;
  4 benches (2,2,2,2): 40320 ways.
If the benches are not linearly ordered the number of ways is A088026.
If we seat an odd number of people on linearly ordered benches the number of ways is A005443. (End)

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000680, A088026.

Table[2^n (2n)!,{n,0,15}] (* Harvey P. Dale, Nov 28 2011 *)

{ for (n=0, 100, write("b065140.txt", n, " ", 2^n*(2*n)!) ) } \\ Harry J. Smith, Oct 11 2009

Hypergeometric generating function, in Maple notation: 1/sqrt(1-8*x), i.e., a(0)=1, a(n)=round(evalf(subs(x=0, n!*diff(1/(sqrt(1-8*x)), x$n)))), for n>=1.
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x>=0} x^n*exp(-sqrt(x/2))/(2*sqrt(2*x)) dx, for n>=0.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*x*(k+1)*(2*k+1)/(4*x*(k+1)*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)/2).
Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)/2). (End)
From Alexandre Herrera, Apr 18 2025: (Start)
Sum_{n>=0} x^(4*n)*(-1)^(n)/a(2n) = cos(x/2)*cosh(x/2).
Sum_{n>=0} x^(4*n+2)*(-1)^(n)/a(2n+1) = sin(x/2)*sinh(x/2).
Sum_{n>=0} x^(2*n)*(-1)^(n)/a(n) = cos(x*sqrt(2)/2).
Sum_{n>=0} x^(2*n)/a(n) = cosh(x*sqrt(2)/2). (End)

A177283 Number of permutations of 2 copies of 1..n with all adjacent differences <= 2 in absolute value.

Original entry on oeis.org

1, 1, 6, 90, 660, 3846, 24326, 148146, 850920, 4788114, 26548704, 144707622, 778116870, 4140736398, 21836757474, 114262071558, 593931441790, 3069689137294, 15786994900896, 80840741811238, 412406022128206, 2096937750144962, 10631278302356284, 53761972675284874

Offset: 0

R. H. Hardin, May 06 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000680.

a(n) = (2n)!/2^n for n<=3.

Terms a(16) and beyond from Andrew Howroyd, May 14 2020

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} :   F_{0}    F_{1}    F_{2}    F_{3}    F_{4}
x = -1:  A165326  A155585  A002105  A292609  A292607
x =  1:  A000012  A000142  A000680  A014606  A014608  ... (m*n)!/m!^n
x =  0:    --     A000012  A000364  A002115  A211212  ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

Original entry on oeis.org

1, 1, 1, 6, 1, 30, 90, 1, 56, 70, 1260, 2520, 1, 90, 420, 3780, 9450, 75600, 113400, 1, 132, 990, 924, 8910, 83160, 34650, 332640, 1247400, 6237000, 7484400, 1, 182, 2002, 6006, 18018, 270270, 252252, 630630, 1081080, 15135120, 12612600, 37837800, 189189000, 681080400, 681080400

Offset: 0

Peter Luschny, Aug 27 2019

We call an irregular triangle T a partition triangle if T(n, k) is defined for n >= 0 and 0 <= k < A000041(n).

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 2. For instance 2*P(4, .) = [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   6]
[3] [1;  30;  90]
[4] [1;  56,  70; 1260; 2520]
[5] [1;  90, 420; 3780, 9450; 75600; 113400]
[6] [1; 132, 990,  924; 8910, 83160,  34650; 332640, 1247400; 6237000; 7484400]
.
T(4, 1) = 56 because [6, 2] is the integer partition 2*P(4, 1) in the canonical order and there are 28 set partitions which have the shape [6, 2] (an example is {{1, 3, 4, 5, 6, 8}, {2, 7}}). Finally, since the order of the sets is taken into account, one gets 2!*28 = 56.

Row sums: A094088, alternating row sums: A028296, main diagonal: A000680, central column A281478, by length: A241171.
Cf. A178803 (m=0), A133314 (m=1), this sequence (m=2), A327023 (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

def GenOrdSetPart(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return [factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes]
def A327022row(n): return GenOrdSetPart(2, n)
for n in (0..6): print(A327022row(n))

A370505 T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i)=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 5, 6, 12, 0, 1, 9, 20, 30, 60, 0, 1, 19, 70, 90, 180, 360, 0, 1, 34, 175, 420, 630, 1260, 2520, 0, 1, 69, 490, 1960, 2520, 5040, 10080, 20160, 0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440, 0, 1, 251, 3948, 21000, 88200, 113400, 226800, 453600, 907200, 1814400

Offset: 0

Alois P. Heinz, Feb 20 2024

			T(0,0) = 1: (only) the empty permutation is 0-dist-increasing.
T(4,2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
Permutation 3142 is 2-dist-increasing and 4-dist-increasing but not 3-dist-increasing.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,    3;
  0, 1,   5,    6,   12;
  0, 1,   9,   20,   30,    60;
  0, 1,  19,   70,   90,   180,   360;
  0, 1,  34,  175,  420,   630,  1260,  2520;
  0, 1,  69,  490, 1960,  2520,  5040, 10080, 20160;
  0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Wikipedia, K-sorted sequence
Wikipedia, Permutation

Columns k=0-2 give: A000007, A057427, A014495.
Row sums give A000142.
Main diagonal gives A001710.
T(2n,n+1) gives A000680 for n>=1.
T(2n,n) gives A370576.
Cf. A248686, A248687, A370506, A370507.

b:= proc(n, k) option remember; `if`(k<1,
     `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
    end:
T:= (n, k)-> b(n, k)-b(n, k-1):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = A248686(n,k) - A248686(n,k-1) for k>=2.

Sum_{k=0..n} (1+n-k) * T(n,k) = A248687(n) for n>=1.

A375222 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that exactly one pair k,k stays at its initial locations 2k-1, 2k.

Original entry on oeis.org

1, 0, 15, 296, 10965, 609864, 47880595, 5047886640, 688359502089, 117929734950320, 24798753695076471, 6280419381186155160, 1885582606127524251805, 662239984799385248609976, 268999138538324585872798395, 125133475474486312764311243744, 66091677106419135401506827779985

Offset: 1

Hugo Pfoertner, Aug 05 2024

nonn

1

			a(3) = 15: The permutations with one stable pair are
  [1, 1, 2, 3, 2, 3], [1, 1, 2, 3, 3, 2], [1, 1, 3, 2, 2, 3], [1, 1, 3, 2, 3, 2],
  [1, 1, 3, 3, 2, 2], [1, 2, 1, 2, 3, 3], [1, 2, 2, 1, 3, 3], [1, 3, 2, 2, 1, 3],
  [1, 3, 2, 2, 3, 1], [2, 1, 1, 2, 3, 3], [2, 1, 2, 1, 3, 3], [2, 2, 1, 1, 3, 3],
  [3, 1, 2, 2, 1, 3], [3, 1, 2, 2, 3, 1], [3, 3, 2, 2, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 1..238

Cf. A000680 (all permutations of this multiset), A375223 (at least one stable pair), A374980.

a375222(n) = {my(p=vector(2*n,i,1+(i-1)\2), m1=0); forperm (p, q, my(m=0); for (k=1, n, if (q[2*k-1]==k && q[2*k]==k, m++)); m1+=(m==1)); m1}

a(n) = n * A374980(n-1). - Alois P. Heinz, Aug 05 2024

a(8) onwards from Alois P. Heinz, Aug 05 2024

A062154 Number T(n,m) of n X m matrices over {0,1,2} with all row and column sums equal to 1 or 2, m=0,..,2*n.

Original entry on oeis.org

1, 0, 2, 1, 0, 1, 13, 18, 6, 0, 0, 18, 189, 450, 360, 90, 0, 0, 6, 450, 4842, 16380, 22140, 12600, 2520, 0, 0, 0, 360, 16380, 190080, 832950, 1631700, 1537200, 680400, 113400, 0, 0, 0, 90, 22140, 832950, 10520010, 56609280, 147533400, 200377800

Offset: 0

Vladeta Jovovic, Jun 06 2001

			Triangle begins:
[0]  1;
[1]  0, 2, 1;
[2]  0, 1, 13, 18, 6;
[3]  0, 0, 18, 189, 450, 360, 90;
[4]  0, 0, 6, 450, 4842, 16380, 22140, 12600, 2520;
[5]  0, 0, 0, 360, 16380, 190080, 832950, 1631700, 1537200, 680400, 113400;
[6]  0, 0, 0, 90, 22140, 832950, 10520010, 56609280, 147533400, 200377800, 144585000, 52390800, 7484400;
T(2, 2)=13, i.e. there are 13 2 X 2 matrices over {0, 1, 2} with all row and column sums equal to 1 or 2: [0 1 / 0 1], [0 1 / 0 2], [0 2 / 1 0], [1 0 / 1 0], [1 1 / 1 1], [1 1 / 2 0], [2 0 / 1 0], [1 1 / 2 0], [1 0 / 2 0], [0 1 / 0 2], [1 1 / 0 1], [1 0 / 1 1], [0 1 / 0 2].

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problem 3.4.15).

Andrew Howroyd, Table of n, a(n) for n = 0..960 (rows 0..30)

Row sums are A062155.
Main diagonal is A062156.
Final terms of each row are A000680.

Row(n)={Vecrev(serlaplace(n!*polcoef((1/sqrt(1-x*y + O(x*x^n))*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2) + O(x*x^n))), n)))}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Feb 03 2021

Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1/sqrt(1-x*y)*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2)).

Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1+(1/2*y^2+2*y)*x+(1/8*y^4+3/2*y^3+13/4*y^2+1/2*y)*x^2+(1/48*y^6+1/2*y^5+25/8*y^4+21/4*y^3+3/2*y^2)*x^3+...

A071798 Number of paths on the surface of the n-dimensional lattice [0..2]^n; i.e., the lattice paths that do not pass through the point (1,1,...,1).

Original entry on oeis.org

0, 2, 54, 1944, 99000, 6966000, 655678800, 80103945600, 12372954249600, 2362712677920000, 547235129437920000, 151247218046601600000, 49191138900262719360000, 18601307697723249058560000, 8093164859945489259936000000, 4014620173473616480790016000000

Offset: 1

T. D. Noe, Jun 06 2002

a(2) + 1 = 3 is prime. a(3) - 1 = 53 is prime. a(5) - 1 = 98999 is prime. a(7) + 1 = 655678801 is prime. a(8) - 1 = 80103945599 is prime, and part of a twin prime, as a(8) + 1 = 80103945601 is prime. a(13) - 1 = 49191138900262719359999 is prime. - Jonathan Vos Post, Sep 01 2009

Alois P. Heinz, Table of n, a(n) for n = 1..238 (first 50 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Lattice Path

Cf. A000680.

Row n=2 of A225094. - Alois P. Heinz, Apr 27 2013

a:= proc(n) option remember; `if`(n<3, (n-1)*n,
       n*((3*n^2-7*n+3)*a(n-1)-(2*n-3)*(n-1)^3*a(n-2))/(n-2))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 26 2013

Mathematica
```
Table[(2n)!/2^n-(n!)^2, {n, 10}]
```

a(n) = (2n)!/2^n - (n!)^2.

More terms from Harvey P. Dale, May 26 2011

A177282 Number of permutations of 2 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 6, 12, 26, 48, 86, 148, 250, 416, 686, 1124, 1834, 2984, 4846, 7860, 12738, 20632, 33406, 54076, 87522, 141640, 229206, 370892, 600146, 971088, 1571286, 2542428, 4113770, 6656256, 10770086, 17426404, 28196554, 45623024, 73819646, 119442740, 193262458

Offset: 0

R. H. Hardin, May 06 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..197 from R. H. Hardin)

Column k=2 of A331562.

Cf. A000680.

a:= proc(n) option remember; `if`(n<4, [1$2, 6, 12][n+1],
     ((8*n-31)*a(n-1) -(4*n-19)*a(n-2) -(3*n-10)*a(n-3)
      +(2*n-10)*a(n-4)) / (3*n-11))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 14 2016

a(n) = (2n)!/2^n = A000680(n) for n<=2.

A210279 (6n)!/6^n.

Original entry on oeis.org

1, 120, 13305600, 29640619008000, 478741050720092160000, 34111736086958726676480000000, 7973107998754741458076119859200000000, 5019026197962676820927435579005599744000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..80

Cf. A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(6*n)/6^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(6 n)!/6^n, {n, 0, 11}] (* Vincenzo Librandi, Feb 15 2013 *)
With[{nn=50},Take[CoefficientList[Series[1/(1-x^6/6),{x,0,nn}],x] Range[0,nn-2]!,{1,-1,6}]] (* Harvey P. Dale, Sep 25 2023 *)

E.g.f.: 1/(1-x^6/6).

cp's OEIS Frontend

A065140 a(n) = 2^n*(2*n)!.

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A177283 Number of permutations of 2 copies of 1..n with all adjacent differences <= 2 in absolute value.

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A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

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A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

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A370505 T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i)=0, 0<=k<=n, read by rows.

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A375222 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that exactly one pair k,k stays at its initial locations 2k-1, 2k.

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A062154 Number T(n,m) of n X m matrices over {0,1,2} with all row and column sums equal to 1 or 2, m=0,..,2*n.

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A071798 Number of paths on the surface of the n-dimensional lattice [0..2]^n; i.e., the lattice paths that do not pass through the point (1,1,...,1).

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A065140 a(n) = 2^n(2n)!.