A000680 -id:A000680 - cp's OEIS frontend

A370576 a(n) is the difference between the number of n-dist-increasing and (n-1)-dist-increasing permutations p of [2n], where p is k-dist-increasing if k>=0 and p(i)

1, 1, 5, 70, 1960, 88200, 5821200, 529729200, 63567504000, 9725828112000, 1847907341280000, 426866595835680000, 117815180450647680000, 38289933646460496000000, 14473594918362067488000000, 6296013789487499357280000000, 3122822839585799681210880000000

Offset: 0

Alois P. Heinz, Feb 22 2024

			a(2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.

Alois P. Heinz, Table of n, a(n) for n = 0..238
Wikipedia, K-sorted sequence
Wikipedia, Permutation

Cf. A000384, A000680, A370505.

a:= n-> ceil((7/9)*(2*n)!/2^n):
seq(a(n), n=0..22);
# second Maple program:
a:= proc(n) a(n):= `if`(n<4, [1$2, 5, 70][n+1], (2*n-1)*n*a(n-1)) end:
seq(a(n), n=0..22);

a(n) = ceiling( (7/9)*(2*n)!/2^n ) = ceiling( (7/9)*A000680(n) ).
a(n) = (2*n-1)*n*a(n-1) for n >= 4 with a(0) = a(1) = 1, a(2) = 5, a(3) = 70.
a(n) = ceiling( (2n)! * [x^(2n)] (7/9)/(1-x^2/2) ).
a(n) = A370505(2n,n).

A382776 Triangle read by rows: T(n,k) is the number of ways to place 2n rooks on a (n+k) X (2n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

Original entry on oeis.org

1, 1, 1, 6, 9, 6, 90, 180, 180, 90, 2520, 6300, 8100, 6300, 2520, 113400, 340200, 529200, 529200, 340200, 113400, 7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400, 681080400, 2724321600, 5658206400, 7858620000, 7858620000, 5658206400, 2724321600, 681080400

Offset: 0

Andrew Howroyd, Apr 04 2025

The configurations are such that k columns will each contain 2 rooks and n-k rows will each contain 2 rooks.

			Triangle begins:
        1;
        1,        1;
        6,        9,        6;
       90,      180,      180,       90;
     2520,     6300,     8100,     6300,     2520;
   113400,   340200,   529200,   529200,   340200,   113400;
  7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400;
  ...
The T(2,0) = 6 configurations are:
  X X . .    X . X .    X . . X    . X X .    . X . X    . . X X
  . . X X    . X . X    . X X .    X . . X    X . X .    X X . .
The T(2,1) = 9 configurations are:
  X X .   X . X   . X X   . . X   . X .   X . .   . . X   . X .   X . .
  . . X   . X .   X . .   X X .   X . X   . X X   . . X   . X .   X . .
  . . X   . X .   X . .   . . X   . X .   X . .   X X .   X . X   . X X

Row sums are A382777.

Column k=0 is A000680.

T(n,k)=binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n)

T(n,k) = binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n).

T(n,n-k) = T(n,k).

A008269 Number of strings on n symbols in Stockhausen problem.

Original entry on oeis.org

1, 2, 9, 112, 2921, 126966, 8204497, 735944084, 87394386417, 13265365173706, 2504688393449081, 575664638548522392, 158222202503521622809, 51242608446417388426622, 19312113111034490954560641, 8379247307752508262094697596, 4146836850351947542340780899937

Offset: 0

Lily Yen

nonn

G. C. Greubel, Table of n, a(n) for n = 0..235
R. C. Read, Combinatorial problems in theory of music, Discrete Math. 167 (1997), 543-551.
Ronald C. Read, Lily Yen, A note on the Stockhausen problem, J. Comb. Theory, Ser. A 76, No. 1, 1-10.

Cf. A000680.

Table[HypergeometricPFQ[{1,1/2,-n},{},-2],{n,0,20}] (* Vaclav Kotesovec, Feb 18 2015 *)

for(n=0,14,print1(2^(-n)*round(intnum(x=0,999,(2+x^2)^n*exp(-x))),", ")) \\ Gerald McGarvey, Oct 12 2007

a(n) = sum(i=0, n, binomial(n,i) * (2*i)!/2^i); \\ Michel Marcus, May 13 2022

a(n) = (2*n^2-5*n+4)*a(n-1) + (-4*n^2+15*n-14)*a(n-2) + (2*n^2-10*n+12)*a(n-3).
a(n) = hypergeom([1, 1/2, -n], [], -2). - Vladeta Jovovic, Apr 08 2007
a(n) = (1/2^n) * Integral_{x>=0} (2+x^2)^n*exp(-x) dx. - Gerald McGarvey, Oct 12 2007
a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Feb 18 2015
a(n) = Sum_{i=0..n} binomial(n,i) * A000680(i). - Håvar Andre Melheim Salbu, May 13 2022

A132911 a(n) = (n+1)(2n)!/2^n.

Original entry on oeis.org

1, 2, 18, 360, 12600, 680400, 52390800, 5448643200, 735566832000, 125046361440000, 26134689540960000, 6585941764321920000, 1969196587532254080000, 689218805636288928000000, 279133616282697015840000000, 129517997955171415349760000000, 68255984922375335889323520000000

Offset: 0

Paul Barry, Sep 04 2007

Define T(n,k)=((1+(-1)^n)/2)*C(k-1+n/2, n/2)*n!/2^(n/2). Then T(n,k) has e.g.f. 1/sum{j=0..k, C(k,j)*(-1)^j*x^(2j)/2^j}. T(n,1) is A000680 with interpolated zeros. T(n,3) is A132912.

Daniele Marchei, Emanuela Merelli, and Andrew Francis, Factorizing the Brauer monoid in polynomial time, arXiv:2402.07874 [math.RA], 2024. See p. 24.

Cf. A000680, A132912.

Table[(n+1) (2n)!/2^n,{n,0,20}] (* Harvey P. Dale, Jun 02 2020 *)

E.g.f.: 1/(1-x^2+x^4/4) (with interpolated zeros);

a(n)-(n+1)*(2*n-1)*a(n-1)=0. - R. J. Mathar, Nov 05 2012

A132912 a(n) = C(n+2,2)(2n)!/2^n.

Original entry on oeis.org

1, 3, 36, 900, 37800, 2381400, 209563200, 24518894400, 3677834160000, 687754987920000, 156808137245760000, 42808621468092480000, 13784376112725778560000, 5169141042272166960000000, 2233068930261576126720000000, 1100902982618957030472960000000

Offset: 0

Paul Barry, Sep 04 2007

Define T(n,k) = ((1+(-1)^n)/2)*C(k-1+n/2, n/2)*n!/2^(n/2). Then T(n,k) has e.g.f. 1/(Sum_{j=0..k} C(k,j)*(-1)^j*x^(2*j)/2^j). T(n,1) is A000680 with interpolated zeros. T(n,2) is A132911.

Cf. A000680, A132911.

Table[(Binomial[n+2,2](2n)!)/2^n,{n,0,20}] (* Harvey P. Dale, Sep 18 2011 *)

E.g.f.: 1/(1-(3/2)*x^2+(3/4)*x^4-(1/8)*x^6) (with interpolated zeros).

a(n) -(n+2)*(2*n-1)*a(n-1)=0. - R. J. Mathar, Nov 05 2012

More terms from Harvey P. Dale, Sep 18 2011

A141765 Triangle T, read by rows, such that row n equals column 0 of matrix power M^n where M is a triangular matrix defined by M(k+m,k) = binomial(k+m,k) for m=0..2 and zeros elsewhere. Width-2-restricted finite functions.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 24, 54, 90, 90, 1, 4, 16, 60, 204, 600, 1440, 2520, 2520, 1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400, 1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400, 1, 7, 49, 336, 2226

Offset: 0

Paul D. Hanna, Jul 28 2008

T(k,n) is the number of distinct ways in which n labeled objects can be distributed in k labeled urns allowing at most 2 objects to fall in each urn. - N-E. Fahssi, Apr 22 2009

T(k,n) is the number of functions f:[n]->[k] such that the preimage set under f of any element of [k] has size 2 or less. - Dennis P. Walsh, Feb 15 2011

			This triangle T begins:
1;
1, 1, 1;
1, 2, 4, 6, 6;
1, 3, 9, 24, 54, 90, 90;
1, 4, 16, 60, 204, 600, 1440, 2520, 2520;
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400;
1, 7, 49, 336, 2226, 14070, 83790, 463680, 2346120, 10636920, 42071400, 139708800, 366735600, 681080400, 681080400,
1, 8, 64, 504, 3864, 28560, 201600, 1345680, 8401680, 48444480, 254016000, 1187524800, 4819953600, 16345929600, 43589145600, 81729648000, 81729648000,
1, 9, 81, 720, 6264, 52920, 430920, 3356640, 24811920, 172504080, 1116536400, 6646147200, 35835307200, 171632260800, 711047937600, 2451889440000, 6620101488000, 12504636144000, 12504636144000,
...
Rows 6 and 8 appear in Park (2015). - _N. J. A. Sloane_, Jan 31 2016
Let M be the triangular matrix that begins:
  1;
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  0,  0,  6,  4,  1;
  0,  0,  0, 10,  5,  1; ...
where M(k+m,k) = C(k+m,k) for m=0,1,2 and zeros elsewhere.
Illustrate that row n of T = column 0 of M^n for n >= 0 as follows.
The matrix square M^2 begins:
   1;
   2,  1;
   4,  4,  1;
   6, 12,  6,  1;
   6, 24, 24,  8,  1;
   0, 30, 60, 40, 10,  1; ...
with column 0 of M^2 forming row 2 of T.
The matrix cube M^3 begins:
   1;
   3,   1;
   9,   6,   1;
  24,  27,   9,   1;
  54,  96,  54,  12,   1;
  90, 270, 240,  90,  15,   1;
  90, 540, 810, 480, 135,  18,   1; ...
with column 0 of M^3 forming row 3 of T.
T(2,3)=6 because there are 6 ways to lodge 3 distinguishable balls, labeled by numbers 1,2 and 3, in 2 distinguishable boxes, each of which can hold at most 2 balls. - _N-E. Fahssi_, Apr 22 2009
T(5,8)=63000 because there are 63000 ways to assign 8 students to a dorm room when there are 5 different two-bed dorm rooms that are available. (See link for details of the count.) - _Dennis P. Walsh_, Feb 15 2011

Dennis Walsh, Width-restricted finite functions
Donghwi Park, Space-state complexity of Korean chess and Chinese chess, arXiv preprint arXiv:1507.06401, 2015

Cf. A003011 (row sums), A000680 (right border); diagonals: A012244, A036774, A003692.

seq(seq(n!*sum(binomial(k,j)*binomial(j,n-j)*2^(j-n),j=ceil(n/2)..k),n=0..2*k),k=1..10); # Dennis P. Walsh, Feb 15 2011

T[k_, n_] := If[n == 0, 1, n! Coefficient[(1 + x + x^2/2)^k, x^n]]; TableForm[Table[T[k, n], {k, 0, 10}, {n, 0, 2 k}]] (* N-E. Fahssi, Apr 22 2009 *)

{T(n,k)=local(M=matrix(n+1,n+1,n,k,if(n>=k,if(n-k<=2,binomial(n-1,k-1))))); if(k>2*n,0,(M^n)[k+1,1])}

T(k,n) = n!*Sum_{i=ceiling(n/2)..k} binomial(k,i)*binomial(i,n-i)*2^(i-n). - Dennis P. Walsh, Feb 15 2011
T(n,2*n) = (2n)!/2^n; thus the rightmost border of T equals A000680.
Main diagonal (central terms) equals A012244.
Other diagonals include A036774 and A003692.
Row sums of triangle T equals A003011, the number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.
T(k,n) = n![x^n](1+x+x^2/2)^k. Double e.g.f.: Sum_{k,n} T(k,n)*(z^k/k!)*(x^n/n!) = exp(z(1+x+x^2/2)). - N-E. Fahssi, Apr 22 2009
T(j+k,n) = Sum_{i=0..n} binomial(n,i)*T(j,i)*T(k,n-i). - Dennis P. Walsh, Feb 15 2011

A177288 Number of permutations of 2 copies of 1..n with all adjacent differences <= 7 in absolute value.

Original entry on oeis.org

1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 7580424852000, 569159625434400, 36636506215016400, 2141931687266957040

Offset: 0

R. H. Hardin, May 06 2010

Cf. A000680.

a(n) = (2n)!/2^n for n<=8.

a(0), a(12) from Alois P. Heinz, May 30 2025

A177289 Number of permutations of 2 copies of 1..n with all adjacent differences <= 8 in absolute value.

Original entry on oeis.org

1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 1522786801536000, 152883132197796000, 13219614969625864800

Offset: 0

R. H. Hardin, May 06 2010

Cf. A000680.

a(n) = (2n)!/2^n for n<=9.

a(0), a(12) from Alois P. Heinz, May 30 2025

A177290 Number of permutations of 2 copies of 1..n with all adjacent differences <= 9 in absolute value.

Original entry on oeis.org

1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 2375880867360000, 367786358267328000, 47683626117839712000

Offset: 0

R. H. Hardin, May 06 2010

Cf. A000680.

a(n) = (2n)!/2^n for n<=10.

a(0)=1 prepended and a(12) added by Alois P. Heinz, Jan 05 2022

A210281 (8n)!/8^n.

Original entry on oeis.org

1, 5040, 326918592000, 1211813284635233280000, 64240926985765022013480960000000, 24899758399899222849902687670779904000000000, 47355329866546908076714664639943599847875543040000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A210280, A210279, A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(8*n)/8^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(8 n)!/8^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^8/8).

cp's OEIS Frontend

A370576 a(n) is the difference between the number of n-dist-increasing and (n-1)-dist-increasing permutations p of [2n], where p is k-dist-increasing if k>=0 and p(i)

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A382776 Triangle read by rows: T(n,k) is the number of ways to place 2*n rooks on a (n+k) X (2*n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

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A008269 Number of strings on n symbols in Stockhausen problem.

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A132911 a(n) = (n+1)*(2*n)!/2^n.

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A132912 a(n) = C(n+2,2)*(2*n)!/2^n.

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A141765 Triangle T, read by rows, such that row n equals column 0 of matrix power M^n where M is a triangular matrix defined by M(k+m,k) = binomial(k+m,k) for m=0..2 and zeros elsewhere. Width-2-restricted finite functions.

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A177288 Number of permutations of 2 copies of 1..n with all adjacent differences <= 7 in absolute value.

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A177289 Number of permutations of 2 copies of 1..n with all adjacent differences <= 8 in absolute value.

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Author

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Extensions

A177290 Number of permutations of 2 copies of 1..n with all adjacent differences <= 9 in absolute value.

A382776 Triangle read by rows: T(n,k) is the number of ways to place 2n rooks on a (n+k) X (2n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

A132911 a(n) = (n+1)(2n)!/2^n.

A132912 a(n) = C(n+2,2)(2n)!/2^n.