A278990 -id:A278990 - cp's OEIS frontend

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.
Row n has 2n+1 entries.
A001498 has a b-file.

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144299, A278990.
Row sums give A001515, column sums A000085.
Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
See A144385 for a generalization.

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
Sum_{k=0..2*n} b(n, k) = A001515(n).
Sum_{n >= 0} b(n, k) = A000085(k).
From G. C. Greubel, Oct 04 2023: (Start)
T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.
Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)

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

A306386 Number of chord diagrams with n chords all having arc length at least 3.

Original entry on oeis.org

1, 0, 0, 1, 7, 68, 837, 11863, 189503, 3377341, 66564396, 1439304777, 33902511983, 864514417843, 23735220814661, 698226455579492, 21914096529153695, 731009183350476805, 25829581529376423945, 963786767538027630275, 37871891147795243899204, 1563295398737378236910447

Offset: 0

Gus Wiseman, Feb 26 2019

nonn

A cyclical form of A190823.

Also the number of 2-uniform set partitions of {1...2n} such that, when the vertices are arranged uniformly around a circle, no block has its two vertices separated by an arc length of less than 3.

			The a(8) = 7 2-uniform set partitions with all arc lengths at least 3:
  {{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}}
  {{1,6},{2,5},{3,7},{4,8}}
  {{1,6},{2,5},{3,8},{4,7}}

Alois P. Heinz, Table of n, a(n) for n = 0..404
Gus Wiseman, The a(5) = 68 chord diagrams with all arc lengths at least 3.

Cf. A000296, A000699, A001006, A001147, A001610, A003436, A038041, A054726, A135042, A170941, A190823, A278990, A306419, A322402, A324011, A324169.

Column k=3 of A324428.

a:= proc(n) option remember; `if`(n<8, [1, 0$2, 1, 7, 68, 837, 11863][n+1],
      ((8*n^4-64*n^3+142*n^2-66*n+109)    *a(n-1)
      -(24*n^4-248*n^3+870*n^2-1106*n+241)*a(n-2)
      +(24*n^4-264*n^3+982*n^2-1270*n+145)*a(n-3)
      -(8*n^4-96*n^3+374*n^2-486*n+33)    *a(n-4)
      -(4*n^3-24*n^2+39*n-2)              *a(n-5))/(4*n^3-36*n^2+99*n-69))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 27 2019

dtui[{},]:={{}};dtui[set:{i,___},n_]:=Join@@Function[s,Prepend[#,s]&/@dtui[Complement[set,s],n]]/@Table[{i,j},{j,Switch[i,1,Select[set,3<#i+2&]]}];
Table[Length[dtui[Range[n],n]],{n,0,12,2}]

a(n) is even <=> n in { A135042 }. - Alois P. Heinz, Feb 27 2019

a(10)-a(16) from Alois P. Heinz, Feb 26 2019

a(17)-a(21) from Alois P. Heinz, Feb 27 2019

A291843 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 0, 1, 5, 3, 36, 33, 2, 329, 388, 72, 3655, 5101, 1545, 64, 47844, 75444, 30700, 3023, 20, 721315, 1248911, 621937, 97200, 3134, 12310199, 22964112, 13269140, 2793713, 180936, 1656, 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352, 4939227215, 10316541393, 7336995966, 2239771686, 293933437, 13977294, 140660

Offset: 0

Gheorghe Coserea, Oct 23 2017

Row n > 0 contains floor((2*n+1)/3) terms.

			A(x;t) = 1 + x^2 + (5 + 3*t)*x^3 + (36 + 33*t + 2*t^2)*x^4 + ...
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]     [6]
[0]  1;
[1]  0;
[2]  1;
[3]  5,         3;
[4]  36,        33,        2;
[5]  329,       388,       72;
[6]  3655,      5101,      1545,      64;
[7]  47844,     75444,     30700,     3023,     20;
[8]  721315,    1248911,   621937,    97200,    3134;
[9]  12310199,  22964112,  13269140,  2793713,  180936,  1656;
[10] 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352;
[11] ...

Gheorghe Coserea, Rows n = 0..123, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A082582, A267827, A278990, A286795, A286798, A286800.

nmax = 11; Clear[Z, Zp]; Z[_] = 0;
Do[
Zp[t_] = Z'[t] + O[t]^n // Normal;
Z[t_] = (-(1/(2L t (1+t)))) (-1 + t - 2L t + 2L^2 t^4 (1 + Zp[t]) + t^2 (1 + 2L + 2L Zp[t]) + L t^3 (3 + 2L + 2(1+L) Zp[t]) + Sqrt[4L t (1+t) (1 + L t)(-1 + t + 2L t^2 + 2(-1 + L) t^2 Zp[t]) + (-1 + t (1 + t + L (-2 + t (2 + t (3 + 2L (1+t))))) + 2L t^2 (1+t)(1 + L t) Zp[t])^2]) + O[t]^n // Normal // Simplify,
{n, nmax+1}];
CoefficientList[#, L]& /@ CoefficientList[Z[t], t] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 23 2018 *)

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1;); y;
};
concat(apply(p->if(p === Pol(0,'t), [0], Vecrev(p)), Vec(A291843_ser(12))))
\\ test: y=A291843_ser(56); 2*x^2*deriv(y,x) == (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x)

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 2*x^2*deriv(y,x) = (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x), with y(0;t)=1, where P_n(t) = Sum_{k=0..floor((2*n-2)/3)} T(n,k)*t^k for n > 0. (see eqn. (24) in Molinari link)
A278990(n) = P_n(0), A294166(n) = P_n(1), A082582(n) = P_n(-1) for n > 1.
A267827(n) = T(3*n+1, 2*n), n > 0. - Danny Rorabaugh, Nov 10 2017

A308356 A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, -1, 1, 0, 1, 5, 5, 0, 1, 0, -1, 36, -120, 15, -1, 1, 0, 1, 329, 6286, 2380, 56, 0, 1, 0, -1, 3655, -557991, 1056496, -52556, 203, -1, 1, 0, 1, 47844, 74741031, 1006985994, 197741887, 1192625, 757, 0, 1

Offset: 0

Seiichi Manyama, May 21 2019

			For (n,k) = (3,2), (1/2) * (Sum_{i=1..3} x^i/i!)^2 = (1/2) * (x + x^2/2 + x^3/6)^2 = (-x)^2/2 + (-3)*(-x)^3/6 + 7*(-x)^4/24 + (-10)*(-x)^5/120 + 10*(-x)^6/720. So A(3,2) = 1 - 3 + 7 - 10 + 10 = 5.
Square array begins:
   1,  0,   0,       0,           0,                0, ...
   1, -1,   1,      -1,           1,               -1, ...
   1,  0,   1,       5,          36,              329, ...
   1, -1,   5,    -120,        6286,          -557991, ...
   1,  0,  15,    2380,     1056496,       1006985994, ...
   1, -1,  56,  -52556,   197741887,   -2063348839223, ...
   1,  0, 203, 1192625, 38987482590, 4546553764660831, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0..4 give A000012, (-1)*A000035, A307349, (-1)*A307350, A307351.
Rows n=0..5 give A000007, A033999, A278990, A308363, A308389, A308390.
Main diagonal gives A308327.
Cf. A144510.

A(n,k) = Sum_{i=k..k*n} b(i) where Sum_{i=k..k*n} b(i) * (-x)^i/i! = (1/k!) * (Sum_{i=1..n} x^i/i!)^k.

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

A278991 a(n) is the number of simple linear diagrams with n+1 chords.

Original entry on oeis.org

0, 1, 3, 24, 211, 2325, 30198, 452809, 7695777, 146193678, 3069668575, 70595504859, 1764755571192, 47645601726541, 1381657584006399, 42829752879449400, 1413337528735664887, 49465522112961344241, 1830184115528550306438, 71375848864779552073957

Offset: 0

N. J. A. Sloane, Dec 07 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..301
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint 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.

Cf. A003436, A003437, A007474, A278990, A278992, A278993, A278994.

a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := a[n] = (2 n - 1) a[n - 1] + (4 n - 3) a[n - 2] + (2 n - 4) a[n - 3]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 10 2016 *)

seq(N) = {
  my(a = vector(N)); a[1]=1; a[2]=3; a[3]=24;
  for (n=4, N, a[n] = (2*n-1)*a[n-1] + (4*n-3)*a[n-2] + (2*n-4)*a[n-3]);
  concat(0, a);
};
seq(20) \\ Gheorghe Coserea, Dec 10 2016

N = 20; x = 'x + O('x^N);
concat(0, Vec(serlaplace((1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x))))) \\ Gheorghe Coserea, Dec 10 2016

E.g.f.: (1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x)).
a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016
a(n) = (2*n-1)*a(n-1) + (4*n-3)*a(n-2) + (2*n-4)*a(n-3). - Gheorghe Coserea, Dec 10 2016

Offset corrected by Gheorghe Coserea, Dec 10 2016

A321666 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.

Original entry on oeis.org

1, 1, 1, 29, 94376, 66218360625, 16985819072511102549, 2421032324142610480402567434373, 271259741131895052775392614041761701799270286, 32119646666355552112999645991677870426882424139287301894021793

Offset: 0

Seiichi Manyama, Nov 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Main diagonal of A322013.

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A278990, A321634.

{a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, (-1)^(n-j)*binomial(n-1, j-1)*x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 27 2019

a(n) = A321634(n)/n!.

a(n) ~ exp(5/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 24 2018

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

A278992 Number of simple chord-labeled chord diagrams with n chords.

Original entry on oeis.org

0, 1, 1, 21, 168, 1968, 26094, 398653, 6872377, 132050271, 2798695656, 64866063276, 1632224748984, 44316286165297, 1291392786926821, 40202651019430461, 1331640833909877144, 46762037794122159492, 1735328399106396110310, 67858430028772637693845

Offset: 1

N. J. A. Sloane, Dec 07 2016

nonn

E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint 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.

Cf. A003436, A003437, A007474, A278990, A278991, A278993, A278994.

terms = 20;
CoefficientList[(Sqrt[1 - 2t]+1)(1/Sqrt[1 - 2t])*E^(Sqrt[1 - 2t] - t - 1) - (2-t)/E^t + O[t]^(terms+1), t]*Range[0, terms]! // Rest (* Jean-François Alcover, Sep 14 2018 *)

E.g.f.: (1+sqrt(1-2*t))*(1-2*t)^(-1/2)*exp(-1-t+sqrt(1-2*t))-(2-t)*exp(-t).
a(n) ~ 2^(n+1/2) * n^n / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016
Conjecture D-finite with recurrence: +(-n+2)*a(n) +(2*n^2-8*n+7)*a(n-1) +(6*n^2-18*n+11)*a(n-2) +(n-1)*(6*n-11)*a(n-3) +2*(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jan 27 2020

cp's OEIS Frontend

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

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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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A306386 Number of chord diagrams with n chords all having arc length at least 3.

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A291843 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A308356 A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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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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A278991 a(n) is the number of simple linear diagrams with n+1 chords.

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