A293157 -id:A293157 - 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

A293881 Number T(n,k) of linear chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 4, 1, 0, 69, 26, 9, 1, 0, 616, 230, 79, 19, 1, 0, 6740, 2509, 854, 252, 39, 1, 0, 87291, 32422, 11105, 3441, 796, 79, 1, 0, 1305710, 484180, 167273, 52938, 14296, 2468, 159, 1, 0, 22149226, 8203519, 2855096, 919077, 265103, 59520, 7564, 319, 1

Offset: 0

Alois P. Heinz, Oct 18 2017

Conjecture: column k>0 is asymptotic to (exp(-k+1) - exp(-k)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       2,      1;
  0,      10,      4,      1;
  0,      69,     26,      9,     1;
  0,     616,    230,     79,    19,     1;
  0,    6740,   2509,    854,   252,    39,    1;
  0,   87291,  32422,  11105,  3441,   796,   79,   1;
  0, 1305710, 484180, 167273, 52938, 14296, 2468, 159,  1;
  ...

Columns k=0-10 give: A000007, A293914, A293915, A293916, A293917, A293918, A293919, A293920, A293921, A293922, A293923.
Row sums give A001147.
T(2n,n) gives A290688.
Main diagonal and first lower diagonal give: A000012, A054135 (for n>0).
Cf. A293157, A293961.

A324428 Number T(n,k) of labeled cyclic chord diagrams with n chords such that every chord has length at least k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 15, 4, 1, 105, 31, 7, 1, 945, 293, 68, 11, 1, 10395, 3326, 837, 159, 18, 1, 135135, 44189, 11863, 2488, 381, 29, 1, 2027025, 673471, 189503, 43169, 7601, 879, 47, 1, 34459425, 11588884, 3377341, 822113, 160784, 23559, 2049, 76, 1, 654729075, 222304897, 66564396, 17066007, 3621067, 607897, 72989, 4788, 123, 1

Offset: 1

Alois P. Heinz, Feb 27 2019

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 1 <= k <= n. T(n,0) = A001147(n), T(0,k) = 1, T(n,k) = 0 for k > n > 0.

			Triangle T(n,k) begins:
        1;
        3,      1;
       15,      4,      1;
      105,     31,      7,     1;
      945,    293,     68,    11,    1;
    10395,   3326,    837,   159,   18,   1;
   135135,  44189,  11863,  2488,  381,  29,  1;
  2027025, 673471, 189503, 43169, 7601, 879, 47, 1;
  ...

Alois P. Heinz, Rows n = 1..19, flattened

Columns k=1-10 give: A001147, A003436, A306386, A324430, A324431, A324432, A324433, A324434, A324435, A324436.
T(n,n-1) gives A000204.
Cf. A293157, A293881, A324429.

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n `if`(n=0 or k<2, doublefactorial(2*n-1),
              b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):
seq(seq(T(n, k), k=1..n), n=1..10);

b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m + Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]],{i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]] + b[n - 1, f, m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];
T[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k-1}], 0, Table[0, {k-1}], k-1]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

T(n,k) = Sum_{j=k..n} A324429(n,j).

A324429 Number T(n,k) of labeled cyclic chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 11, 3, 1, 0, 74, 24, 6, 1, 0, 652, 225, 57, 10, 1, 0, 7069, 2489, 678, 141, 17, 1, 0, 90946, 32326, 9375, 2107, 352, 28, 1, 0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1, 0, 22870541, 8211543, 2555228, 661329, 137225, 21510, 1973, 75, 1

Offset: 0

Alois P. Heinz, Feb 27 2019

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 0 <= k <= n. T(n,k) = 0 for k > n.

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       2,      1;
  0,      11,      3,      1;
  0,      74,     24,      6,     1;
  0,     652,    225,     57,    10,    1;
  0,    7069,   2489,    678,   141,   17,   1;
  0,   90946,  32326,   9375,  2107,  352,  28,  1;
  0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1;
  ...

Alois P. Heinz, Rows n = 0..19, flattened

Columns k=0-10 give: A000007, A324445, A324446, A324447, A324448, A324449, A324450, A324451, A324452, A324453, A324454.
Row sums give A001147.
Main diagonal gives A000012.
T(n+1,n) gives A001610.
Cf. A293157, A293881, A324428.

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n `if`(n=0 or k<2, doublefactorial(2*n-1),
              b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):
T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m +  Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]], {i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j-1]]] + b[n-1, f, m + l[[1]], Append[ReplacePart[ l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];
A[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k - 1}], 0, Table[0, {k - 1}], k - 1]];
T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

T(n,k) = A324428(n,k) - A324428(n,k+1) for k > 0, T(n,0) = A000007(n).

A190823 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 2.

Original entry on oeis.org

1, 0, 0, 1, 10, 99, 1146, 15422, 237135, 4106680, 79154927, 1681383864, 39034539488, 983466451011, 26728184505750, 779476074425297, 24281301468714902, 804688068731837874, 28269541494090294129, 1049450257149017422000, 41050171013933837206545

Offset: 0

R. H. Hardin, May 21 2011

From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} such that no block has its two vertices differing by less than 3. For example, the a(4) = 10 set partitions are:
  {{1,4}, {2,6}, {3,7}, {5,8}}
  {{1,4}, {2,7}, {3,6}, {5,8}}
  {{1,5}, {2,6}, {3,7}, {4,8}}
  {{1,5}, {2,6}, {3,8}, {4,7}}
  {{1,5}, {2,7}, {3,6}, {4,8}}
  {{1,5}, {2,8}, {3,6}, {4,7}}
  {{1,6}, {2,5}, {3,7}, {4,8}}
  {{1,6}, {2,5}, {3,8}, {4,7}}
  {{1,7}, {2,5}, {3,6}, {4,8}}
  {{1,8}, {2,5}, {3,6}, {4,7}}
(End)

			All solutions for n=4 (read downwards):
  1    1    1    1    1    1    1    1    1    1
  2    2    2    2    2    2    2    2    2    2
  3    3    3    3    3    3    3    3    3    3
  4    4    4    4    1    4    4    1    4    4
  1    1    2    1    4    2    1    4    2    2
  3    3    1    2    2    3    2    3    1    3
  2    4    4    4    3    4    3    2    3    1
  4    2    3    3    4    1    4    4    4    4

Alois P. Heinz, Table of n, a(n) for n = 0..404
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

Distance of 1 instead of 2 gives |A000806|.
Column k=3 of A293157.
Cf. A000699, A001147 (2-uniform set partitions), A003436, A005493, A011968, A170941, A278990 (distance 2+ version), A306386 (cyclical version).

I:=[1,0,0,1,10,99]; [n le 5 select I[n] else 2*n*Self(n-1) -2*(3*n-8)*Self(n-2) +2*(3*n-11)*Self(n-3) -2*(n-5)*Self(n-4) -Self(n-5): n in [1..40]]; // G. C. Greubel, Dec 03 2023

a[0]=1; a[1]=0; a[2]=0; a[3]=1; a[4]=10; a[5]=99; a[n_] := a[n] = (2*n+2) a[n-1] - (6*n-10) a[n-2] + (6*n-16) a[n-3] - (2*n-8) a[n-4] - a[n-5]; Array[a, 20, 0] (* based on Sullivan's formula, Giovanni Resta, Mar 20 2017 *)
dtui[{}]:={{}};dtui[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@dtui[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+2&]}];
Table[Length[dtui[Range[n]]],{n,0,12,2}] (* Gus Wiseman, Feb 27 2019 *)

@CachedFunction
def a(n): # a = A190823
    if (n<6): return (1,0,0,1,10,99)[n]
    else: return 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Dec 03 2023

a(n) = 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5) (proved). - Everett Sullivan, Mar 16 2017

a(n) ~ 2^(n+1/2) * n^n / exp(n+2), based on Sullivan's formula. - Vaclav Kotesovec, Mar 21 2017

a(16)-a(20) (using Everett Sullivan's formula) from Giovanni Resta, Mar 20 2017

a(0)=1 prepended by Alois P. Heinz, Oct 17 2017

A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 4, 6, 5, 0, 27, 36, 28, 14, 0, 248, 310, 225, 120, 42, 0, 2830, 3396, 2332, 1210, 495, 132, 0, 38232, 44604, 29302, 14560, 6006, 2002, 429, 0, 593859, 678696, 430200, 204540, 81900, 28392, 8008, 1430, 0, 10401712, 11701926, 7204821, 3289296, 1263780, 431256, 129948, 31824, 4862

Offset: 0

R. J. Mathar, Dec 06 2018

If all subsets are allowed instead of just pairs (chords), we get A324173. The rightmost column is A000108 (see Riordan). - Gus Wiseman, Feb 27 2019

			From _Gus Wiseman_, Feb 27 2019: (Start)
Triangle begins:
  1
  0      1
  0      1      2
  0      4      6      5
  0     27     36     28     14
  0    248    310    225    120     42
  0   2830   3396   2332   1210    495    132
  0  38232  44604  29302  14560   6006   2002    429
  0 593859 678696 430200 204540  81900  28392   8008   1430
Row n = 3 counts the following chord diagrams (see link for pictures):
  {{1,3},{2,5},{4,6}}  {{1,2},{3,5},{4,6}}  {{1,2},{3,4},{5,6}}
  {{1,4},{2,5},{3,6}}  {{1,3},{2,4},{5,6}}  {{1,2},{3,6},{4,5}}
  {{1,4},{2,6},{3,5}}  {{1,3},{2,6},{4,5}}  {{1,4},{2,3},{5,6}}
  {{1,5},{2,4},{3,6}}  {{1,5},{2,3},{4,6}}  {{1,6},{2,3},{4,5}}
                       {{1,5},{2,6},{3,4}}  {{1,6},{2,5},{3,4}}
                       {{1,6},{2,4},{3,5}}
(End)

P. Flajolet and M. Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, p 191-201, eq (2)
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Gus Wiseman, Chords diagrams with 3 chords, organized by number of components.

Cf. A000699 (k = 1 column), A001147 (row sums), A000108 (diagonal), A002694 (subdiagonal k = n - 1).

Cf. A000096, A003436, A016098, A099947, A136653, A278990, A293157, A324173, A324323, A324327, A324328.

The g.f. satisfies g(z,w) = 1+w*A000699(w*g^2), where A000699(z) is the g.f. of A000699.

Offset changed to 0 by Gus Wiseman, Feb 27 2019

A293156 Number of linear chord diagrams with n+2 chords such that every chord has length at least n.

Original entry on oeis.org

15, 36, 99, 292, 876, 2628, 7884, 23652, 70956, 212868, 638604, 1915812, 5747436, 17242308, 51726924, 155180772, 465542316, 1396626948, 4189880844, 12569642532, 37708927596, 113126782788, 339380348364, 1018141045092, 3054423135276, 9163269405828, 27489808217484

Offset: 1

N. J. A. Sloane, Oct 10 2017

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (3).

A diagonal of A293157.

Join[{15,36,99},NestList[3#&,292,30]] (* Harvey P. Dale, Sep 25 2018 *)

Vec(x*(15 - 9*x - 9*x^2 - 5*x^3) / (1 - 3*x) + O(x^30)) \\ Colin Barker, Oct 18 2017

G.f.: (5*x^3+9*x^2+9*x-15)*x/(3*x-1). - Alois P. Heinz, Oct 17 2017
From Colin Barker, Oct 18 2017: (Start)
a(n) = 292*3^(n-4) for n>3.
a(n) = 3*a(n-1) for n>4.
(End)

More terms from Alois P. Heinz, Oct 17 2017

A190824 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 3.

Original entry on oeis.org

1, 0, 0, 0, 1, 20, 292, 4317, 69862, 1251584, 24728326, 535333713, 12616277612, 321762301156, 8833356675295, 259803215904436, 8151872288855008, 271848098526643604, 9602477503845334715, 358185069617609239664, 14070369448248794118128, 580623906507508489287367

Offset: 0

R. H. Hardin, May 21 2011

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..404
Vaclav Kotesovec, Recurrence (of order 12)
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

Column k=4 of A293157.

a(n) ~ 2^(n + 1/2) * n^n / exp(n+3). - Vaclav Kotesovec, Oct 26 2017

a(0)=1 prepended and a(15)-a(21) added by Alois P. Heinz, Oct 17 2017

A190825 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 40, 876, 16924, 332507, 6944594, 156127796, 3783620060, 98614428186, 2754950165519, 82200735013648, 2610496155216456, 87952061484214504, 3134296573781405311, 117816169705165137052, 4659486136831587519216, 193431371632142599974128

Offset: 0

R. H. Hardin, May 21 2011

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..404
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

Column k=5 of A293157.

a(n) ~ 2^(n + 1/2) * n^n / exp(n+4). - Vaclav Kotesovec, Oct 26 2017

a(0), a(18)-a(21) from Alois P. Heinz, Oct 17 2017

cp's OEIS Frontend

A278990 Number of loopless linear chord diagrams with n chords.

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A293881 Number T(n,k) of linear chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A324428 Number T(n,k) of labeled cyclic chord diagrams with n chords such that every chord has length at least k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A324429 Number T(n,k) of labeled cyclic chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A190823 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 2.

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A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.

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A293156 Number of linear chord diagrams with n+2 chords such that every chord has length at least n.

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