A322013 -id:A322013 - cp's OEIS frontend

A278990 Number of loopless linear chord diagrams with n chords.

1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411

Offset: 0

N. J. A. Sloane, Dec 07 2016

nonn

See the signed version of these numbers, A000806, for much more information about these numbers.
From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:
  {{1,3},{2,5},{4,6}}
  {{1,4},{2,5},{3,6}}
  {{1,4},{2,6},{3,5}}
  {{1,5},{2,4},{3,6}}
  {{1,6},{2,4},{3,5}}
(End)
From Gus Wiseman, Jul 05 2020: (Start)
Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.
  (1,2,3,1,2,3)
  (1,2,3,1,3,2)
  (1,2,3,2,1,3)
  (1,2,3,2,3,1)
  (1,2,1,3,2,3)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..404 (terms 0..200 from Gheorghe Coserea)
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
E. Krasko, I. Labutin, and A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218 [math.CO], 2017, Table 1.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
Gus Wiseman, The a(4) = 36 loopless linear chord diagrams.
Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Column k=0 of A079267.
Column k=2 of A293157.
Row n=2 of A322013.
Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.
Anti-run compositions are A003242.
Separable partitions are A325534.
Cf. A007716, A292884, A333489.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.

[n le 2 select 2-n else (2*n-3)*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 26 2023

RecurrenceTable[{a[n]== (2n-1)a[n-1] +a[n-2], a[0]==1, a[1]==0}, a, {n,0,20}] (* Vaclav Kotesovec, Sep 15 2017 *)
FullSimplify[Table[-I*(BesselI[1/2+n,-1] BesselK[3/2,1] - BesselI[3/2,-1] BesselK[1/2+ n,1]), {n,0,20}]] (* Vaclav Kotesovec, Sep 15 2017 *)
Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* Eric W. Weisstein, Nov 14 2018 *)
Table[Sqrt[2/Pi]/E ((-1)^n Pi BesselI[1/2+n,1] +BesselK[1/2+n,1]), {n,0,20}] // FunctionExpand // FullSimplify (* Eric W. Weisstein, Nov 14 2018 *)
twouniflin[{}]:={{}};twouniflin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

seq(N) = {
  my(a = vector(N)); a[1] = 0; a[2] = 1;
  for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
  concat(1, a);
};
seq(20) \\ Gheorghe Coserea, Dec 09 2016

def A278990_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A278990_list(30) # G. C. Greubel, Sep 26 2023

From Gheorghe Coserea, Dec 09 2016: (Start)
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.
E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.
a(n) - a(n-1) = A003436(n) for all n >= 2. (End)
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) = sqrt(2)*exp(-1)*(BesselK(1/2 + n, 1)/sqrt(Pi) - i*sqrt(Pi)*BesselI(1/2 + n, -1)), where i is the imaginary unit.
a(n) ~ 2^(n+1/2) * n^n  / exp(n+1). (End)
a(n) = A114938(n)/n! - Gus Wiseman, Jul 05 2020 (from Alexander Burstein's formula at A114938).
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).
G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).
E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x). (End)

a(0)=1 prepended by Gheorghe Coserea, Dec 09 2016

A322093 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 2, 0, 6, 2, 0, 24, 30, 2, 0, 120, 864, 174, 2, 0, 720, 39480, 41304, 1092, 2, 0, 5040, 2631600, 19606320, 2265024, 7188, 2, 0, 40320, 241133760, 16438575600, 11804626080, 134631576, 48852, 2, 0, 362880, 29083420800, 22278418248240, 131402141197200, 7946203275000, 8437796016, 339720, 2, 0

Offset: 1

Seiichi Manyama, Nov 26 2018

			Square array begins:
   1, 2,    6,        24,           120,                 720, ...
   0, 2,   30,       864,         39480,             2631600, ...
   0, 2,  174,     41304,      19606320,         16438575600, ...
   0, 2, 1092,   2265024,   11804626080,     131402141197200, ...
   0, 2, 7188, 134631576, 7946203275000, 1210527140790855600, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Columns k=3 gives A110706.
Rows n=1..10 give A000142, A114938, A193638, A321633, A322126, A321382, A322095, A322096, A322145, A322146.
Main diagonal gives A321634.
Cf. A322013.

Table[Table[SeriesCoefficient[1/(1 - Sum[x[i]/(1 + x[i]), {i, 1, n}]), Sequence @@ Table[{x[i], 0, k}, {i, 1, n}]],{n, 1, 6}], {k, 1, 5}] (* Zlatko Damijanic, Nov 03 2024 *)

q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n,k) = subst(serlaplace(q(n,x)^k), x, 1) \\ Andrew Howroyd, Feb 03 2024

A(n,k) = k! * A322013(n,k).
Let q_n(x) = Sum_{i=1..n} (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!.
A(n,k) = Integral_{0..infinity} (q_n(x))^k * exp(-x) dx.

A190917 Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 1, 5, 29, 182, 1198, 8142, 56620, 400598, 2872754, 20824778, 152303410, 1122149800, 8319825040, 62017475600, 464452683432, 3492568119566, 26358270711370, 199565061455634, 1515311001158482, 11535716330003876, 88025068713285476, 673124069796140900

Offset: 0

R. H. Hardin, May 23 2011

nonn

			All solutions for n=2:
  1    1    1    1    1
  2    2    2    2    2
  3    3    3    3    1
  1    2    2    1    3
  3    3    1    2    2
  2    1    3    3    3

Seiichi Manyama, Table of n, a(n) for n = 0..1111 (terms 1..200 from R. H. Hardin)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{3,m}/3!.

Column 3 of A322013.

Cf. A000012 (b=2), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9), A322061 (b=10).

[(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]])/3: n in [1..25]]; // G. C. Greubel, Nov 24 2018

a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1],
      ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 09 2023

Table[(1/3)*Sum[Binomial[n-1, k]*(Binomial[n-1, k]*Binomial[2*n+1-2*k, n+1] + Binomial[n-1, k+1]*Binomial[2*n-2*k, n+1]), {k,0,Floor[n/2]}], {n,1,25}] (* G. C. Greubel, Nov 24 2018 *)

A190917(n) = sum(k=0, n\2, binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1))) / 3; \\ Max Alekseyev, Dec 10 2017

[(1/3)*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1))  for k in range(1+floor(n/2))) for n in (1..25)] # G. C. Greubel, Nov 24 2018

a(n) = A110706(n) / 6 for n >= 1.

n*(n+1)*a(n) - (n+1)*(7*n-4)*a(n-1) - 8*(n-2)^2*a(n-2) = 0. - R. J. Mathar, Nov 01 2015 from A110706

a(0)=1 prepended by Alois P. Heinz, Sep 09 2023

A190826 Number of permutations of 3 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 29, 1721, 163386, 22831355, 4420321081, 1133879136649, 372419001449076, 152466248712342181, 76134462292157828285, 45552714996556390334921, 32173493282909179882613934, 26487410329744429030530295991, 25143126122564855343240882599761, 27260957330891104469298062949026065

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some of the a(3) = 29 solutions for n=3: 123232131, 123121323, 123123213, 123212313, 123213123, 121323132, 123132312, 123123123, 123231213, 121323123, 121321323, 121312323, 121323231, 123231321, 121313232, 123132321, ...

Seiichi Manyama, Table of n, a(n) for n = 0..223 (terms 0..101 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,3}/3!.

Cf. A192990, A193624, A193638.

Row n=3 of A322013.

B:=Binomial;
f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(k-j): k in [Ceiling(j/2)..n]]) >;
A190826:= func< n | (-1/2)^n*(&+[Factorial(j)*B(n+j,j)*f(n,j): j in [0..2*n]]) >;
[A190826(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023

a[n_]:= 1/(6^n*n!)*Sum[(n+j)! Sum[Binomial[n,k] Binomial[2k,j] (-3)^(n+k-j), {k, Ceiling[j/2], n}], {j,0,2n}]; Array[a, 16, 0] (* Jean-François Alcover, Jul 22 2017, after Tani Akinari's code for A193638 *)

b=binomial;
def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(k-j) for k in range((j//2),n+1))
def A190826(n): return (-1/2)^n*sum(factorial(j)*b(n+j,j)*f(j,n) for j in range(2*n+1))
[A190826(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

a(n) = A193624(n)/(6^n * n!), for n >= 1.
a(n) = A193638(n)/n!, for n >= 1.
a(n) = A192990(binomial(n+2,3)) / (6^n * n!), for n >= 1.
2*a(n) -3*(3*n^2-3*n+4)*a(n-1) +2*(9*n^2-42*n+47)*a(n-2) +8*(3*n-7)*a(n-3) -8*a(n-4) = 0. - R. J. Mathar, May 23 2014
a(n) = (1/(6^n * n!)) * Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - Jean-François Alcover, Jul 22 2017
a(n) ~ 3^(2*n + 1/2) * n^(2*n) / (2^n * exp(2*n + 2)). - Vaclav Kotesovec, Nov 24 2018

a(0)=1 prepended by Alois P. Heinz, Jul 22 2017

A190830 Number of permutations of 4 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 182, 94376, 98371884, 182502973885, 551248360550999, 2536823683737613858, 16904301142107043464659, 156690501089429126239232946, 1955972150994131850032960933480, 32016987304767134806200915633253966, 672058204939482014438623912695190927357

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some solutions for n=3:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
  3  1  3  3  1  3  3  3  3  3  3  1  1  3  1  1
  1  2  2  2  2  2  2  1  1  2  1  3  3  1  3  2
  2  3  3  3  1  1  3  2  2  3  2  1  2  3  1  1
  3  2  1  2  3  2  1  3  1  1  3  2  3  2  2  3
  1  3  2  3  2  3  3  1  3  2  2  3  2  1  3  1
  3  2  3  1  3  1  2  3  1  1  1  2  1  3  2  3
  1  1  1  3  1  3  1  2  2  3  3  3  2  2  3  2
  2  3  2  1  3  2  3  1  3  1  1  2  3  3  1  3
  3  1  1  2  2  3  1  3  2  2  2  3  1  1  3  2
  2  3  3  1  3  1  2  2  3  3  3  1  3  2  2  3

Seiichi Manyama, Table of n, a(n) for n = 0..158 (terms 1..28 from R. J. Mathar)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,4}/c!.

Row n=4 of A322013.

Cf. A190826, A321633.

From Vaclav Kotesovec, Nov 24 2018: (Start)
Recurrence: 3*(64*n^3 - 280*n^2 + 414*n - 245)*a(n) = (2048*n^6 - 12032*n^5 + 30400*n^4 - 42608*n^3 + 32484*n^2 - 14624*n + 1731)*a(n-1) + 3*(3840*n^5 - 20640*n^4 + 40104*n^3 - 36340*n^2 + 23378*n - 13429)*a(n-2) - 18*(384*n^4 - 1488*n^3 + 1556*n^2 - 986*n + 649)*a(n-3) - 27*(64*n^3 - 88*n^2 + 46*n - 47)*a(n-4).
a(n) ~ 2^(5*n+1) * n^(3*n) / (3^n * exp(3*n + 3)). (End)

a(0)=1 prepended by Seiichi Manyama, Nov 16 2018

A190833 Number of permutations of 5 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 1198, 5609649, 66218360625, 1681287695542855, 81644850343968535401, 6945222145021508480249929, 967335448974819561548523580438, 209141786137614009701487336108267723

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some solutions for n=3
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....1....3....3....3....3....3....3....3....3....1....1....3....3....1....3
..2....2....2....2....1....2....2....2....1....1....2....3....2....2....3....2
..3....3....3....3....2....1....3....1....2....3....3....2....1....1....2....3
..1....2....1....2....3....2....1....3....3....1....1....1....3....2....3....2
..2....3....3....3....1....3....3....2....2....3....2....3....1....1....1....1
..1....1....1....1....3....1....1....3....1....2....3....2....3....3....2....2
..3....2....2....3....2....3....3....2....3....3....2....3....2....2....3....3
..1....3....1....1....3....1....2....3....2....1....3....1....3....1....1....1
..3....1....3....2....2....2....3....1....3....2....1....3....2....3....2....2
..1....3....2....1....3....3....2....3....1....3....3....2....1....2....3....3
..2....2....1....2....1....1....1....1....2....2....1....3....3....3....1....1
..3....1....2....1....2....2....2....2....1....1....3....2....2....1....2....3
..2....3....3....3....1....3....1....1....3....2....2....1....1....3....3....1

Seiichi Manyama, Table of n, a(n) for n = 0..100 (terms 1..17 from R. J. Mathar)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015)

Row n=5 of A322013.

From Vaclav Kotesovec, Nov 24 2018: (Start)
Recurrence: 72*(2562500*n^6 - 28700000*n^5 + 132339375*n^4 - 325827750*n^3 + 454176325*n^2 - 335170330*n + 94842168)*a(n) = 3*(1601562500*n^10 - 21140625000*n^9 + 123391796875*n^8 - 423007187500*n^7 + 944775218750*n^6 - 1434958662500*n^5 + 1501190434375*n^4 - 1066734651000*n^3 + 489294264300*n^2 - 138373925520*n + 5253272832)*a(n-1) + 40*(2946875000*n^9 - 37425312500*n^8 + 202210281250*n^7 - 613007709375*n^6 + 1158238267500*n^5 - 1429774838250*n^4 + 1130386059700*n^3 - 439029380875*n^2 - 90639409450*n + 124751567448)*a(n-2) - 1440*(166562500*n^8 - 1865500000*n^7 + 8477778125*n^6 - 20572469375*n^5 + 30057234250*n^4 - 27819019325*n^3 + 13875953795*n^2 + 328927290*n - 3329053712)*a(n-3) - 5760*(15375000*n^7 - 133762500*n^6 + 442461250*n^5 - 764293375*n^4 + 806629450*n^3 - 482585405*n^2 + 27997588*n + 122388456)*a(n-4) + 6912*(2562500*n^6 - 13325000*n^5 + 27276875*n^4 - 32220250*n^3 + 22166825*n^2 - 3068430*n - 5777712)*a(n-5).
a(n) ~ 5^(4*n + 1/2) * n^(4*n) / (24^n * exp(4*n + 4)). (End)

a(0)=1 prepended by Seiichi Manyama, Nov 16 2018

A190835 Number of permutations of 6 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 8142, 351574834, 47940557125969, 16985819072511102549, 13519747358522016160671387, 21671513613423101256198918372909, 64311863997340571475504065539218471107, 330586922756304429697714946501284146322953006

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some solutions for n=3
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....3....1....3....1....3....3....3....1....1....3....3....3....3....3....3
..2....2....3....2....3....2....2....1....2....3....1....1....2....2....2....1
..1....3....2....3....2....1....3....3....1....2....3....3....3....3....1....2
..2....1....3....1....3....3....1....2....3....3....2....2....2....1....3....1
..3....2....1....2....1....2....3....3....1....2....3....3....1....2....2....2
..2....1....3....3....3....1....2....2....3....3....1....1....3....3....3....3
..3....2....2....1....2....2....1....3....2....2....2....3....1....2....2....2
..1....3....3....3....3....3....2....1....3....1....1....1....3....1....1....1
..3....1....2....2....2....1....1....3....2....3....2....2....2....3....3....3
..1....2....1....3....1....3....2....1....3....1....1....3....1....2....1....1
..3....3....2....2....3....1....1....2....2....2....3....2....2....1....2....2
..1....1....3....1....2....3....3....3....1....3....2....3....1....3....1....3
..2....3....2....3....1....2....1....1....2....1....1....2....3....1....2....2
..3....1....1....1....2....1....3....2....3....3....3....1....1....2....3....3
..1....3....3....2....3....2....2....1....1....2....2....2....2....1....1....1
..2....2....1....1....1....3....3....2....3....1....3....1....3....3....3....3

Seiichi Manyama, Table of n, a(n) for n = 0..100 (terms 1..14 from R. J. Mathar)

Row n=6 of A322013.

a(n) ~ sqrt(6) * 324^n * n^(5*n) / (5^n * exp(5*n + 5)). - Vaclav Kotesovec, Nov 24 2018

a(0)=1 prepended by Seiichi Manyama, Nov 16 2018

A190918 Number of permutations of n copies of 1..4 introduced in order 1..4 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 36, 1721, 94376, 5609649, 351574834, 22875971289, 1530622143864, 104650147201049, 7279277647839552, 513492654638478897, 36647810215955194122, 2641438793287744496337, 191996676519223794534702, 14057702378132873242943289, 1035863834231020871413190808

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2:
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....3....3....3....3....3....3....1....3....3....1....3....3....3....1....3
..4....1....4....2....4....1....2....3....4....4....3....4....1....2....2....4
..2....4....1....4....1....4....4....4....2....2....4....3....4....4....3....2
..4....2....2....3....3....2....1....3....3....1....2....1....3....1....4....4
..3....3....4....1....4....4....3....4....4....4....3....4....4....4....3....1
..1....4....3....4....2....3....4....2....1....3....4....2....2....3....4....3

Richard Copley, Table of n, a(n) for n = 1..502 (first 106 terms from R. H. Hardin)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{4,m}/4!.

Column k=4 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190920 (b=5), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9).

Conjecture: n^3 *(n-1) *(2*n+1) *(5*n-7) *a(n) -2*(n-1) *(415*n^5 -1026*n^4 +727*n^3 -90*n^2 -98*n +36)*a(n-1) +(1630*n^4 -6387*n^3 +7388*n^2 +111*n -3510) *(n-2)^2 *a(n-2) -162*(3+5*n) *(n-2)^2 *(n-3)^3 *a(n-3)=0. - R. J. Mathar, Nov 01 2015

a(n) ~ 3^(4*n-2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 24 2018

A190920 Number of permutations of n copies of 1..5 introduced in order 1..5 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 329, 163386, 98371884, 66218360625, 47940557125969, 36533294879349056, 28920026907938624194, 23575497690601916022516, 19672658572012343899666292, 16730974132035148942028759656, 14455459908454408519322566567054

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....3....3....3....3....3....3....3....1....3....3....3....3....3....3....3
..4....4....2....1....4....4....4....4....3....4....4....4....1....4....4....4
..1....5....4....4....1....5....3....5....4....5....3....5....4....3....2....5
..5....3....5....2....5....2....4....4....5....2....2....3....5....2....3....2
..2....2....4....3....4....1....5....5....2....4....5....5....4....5....5....5
..5....1....3....5....2....3....1....1....4....3....1....4....2....1....4....4
..3....4....5....4....3....4....5....3....5....5....5....1....5....4....1....3
..4....5....1....5....5....5....2....2....3....1....4....2....3....5....5....1

Seiichi Manyama, Table of n, a(n) for n = 1..334 (terms 1..55 from R. H. Hardin)

Column k=5 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9).

a(n) ~ 25 * sqrt(5) * 2^(10*n-7) / (27 * Pi^2 * n^2). - Vaclav Kotesovec, Nov 24 2018

A190923 Number of permutations of n copies of 1..6 introduced in order 1..6 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 3655, 22831355, 182502973885, 1681287695542855, 16985819072511102549, 183095824753841610373405, 2070756746775910218326948065, 24302858067615766089801166488125, 293736218147318801678882792470437721

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....3....3....3....3....3....3....3....3....3....3....3....3....3....3....3
..4....2....4....4....4....2....4....4....4....4....4....4....4....4....4....4
..5....4....5....5....5....4....1....5....5....2....5....1....5....5....1....3
..1....5....4....3....3....1....3....4....2....5....6....4....6....6....5....4
..4....6....2....2....6....4....5....5....4....6....1....5....3....2....2....2
..5....3....1....6....5....5....6....1....5....5....5....6....2....6....6....5
..2....5....6....5....6....3....5....6....3....3....3....2....6....3....3....1
..6....6....5....6....2....6....4....3....6....4....6....5....4....1....6....6
..3....1....6....1....1....5....2....6....1....1....2....6....5....4....5....5
..6....4....3....4....4....6....6....2....6....6....4....3....1....5....4....6

Seiichi Manyama, Table of n, a(n) for n = 1..240 (terms 1..35 from R. H. Hardin)

Column k=6 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190920 (b=5), A190927 (b=7), A190932 (b=8), A321987 (b=9).

a(n) ~ 9 * 5^(6*n-2) / (128 * sqrt(2) * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 24 2018

cp's OEIS Frontend

A278990 Number of loopless linear chord diagrams with n chords.

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A322093 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k with no element equal to another within a distance of 1.

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A190917 Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.

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A190826 Number of permutations of 3 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A190830 Number of permutations of 4 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A190833 Number of permutations of 5 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A190835 Number of permutations of 6 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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