A000699 -id:A000699 - cp's OEIS frontend

A113672 Self-convolution 6th power equals A113666, where a(n) = n*A113666(n-1) for n>=1, with a(0)=1.

1, 1, 12, 261, 7784, 287145, 12452256, 616408534, 34178166288, 2094929612766, 140568321437700, 10246761825942972, 806426083421461440, 68162575162983744079, 6159817390723312545936, 592796927295190983761100

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Cf. A113666, A000699, A113669, A113670, A113671, A113673, A113674.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^6));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^6],
(2) [x^n] exp( x*A(x)^6 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^6 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113673 Self-convolution 7th power equals A113667, where a(n) = n*A113667(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 14, 357, 12488, 540155, 27453258, 1591997162, 103362754048, 7415833578300, 582246803894350, 49648781879763836, 4569614321483063496, 451606519694514555917, 47709061981854231868308

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Cf. A113667, A000699, A113669, A113670, A113671, A113672, A113674.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^7));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^7],
(2) [x^n] exp( x*A(x)^7 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^7 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113674 Self-convolution 8th power equals A113668, where a(n) = n*A113668(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 16, 468, 18784, 932030, 54321840, 3611129620, 268687287744, 22085224470873, 1986091468594160, 193935237759263880, 20436302307290415264, 2311999369405933686648, 279558778132903394262032

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Cf. A113668, A000699, A113669, A113670, A113671, A113672, A113673.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^8));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^8],
(2) [x^n] exp( x*A(x)^8 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^8 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

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

A280778 Numerators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

Original entry on oeis.org

1, -4, -6, -154, -1610, -34588, -4666292, -553625626, -1158735422, -388434091184, -31268175015478, -2796356409576766, -4624948938397276052, -1691272281281652408568, -2154089954877183990112, -170222948041126582837968646, -5761785676811885455064909606, -55629298859254851627617870836

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -4, -6, -154/3, -1610/3, -34588/5, -4666292/45, -553625626/315, -1158735422/35, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280779, A280780, A280781.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = {
  my(M = subst(x*Ser(A000699_seq(N)), x, x/(1-x)^2));
  Vec(x/(1-x)*exp(1-x/2-(1-x)^2/(2*x)*(2*M + M^2))/M);
};
apply(numerator, seq(18))  \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280779 Denominators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 45, 315, 35, 567, 2025, 7425, 467775, 6081075, 257985, 638512875, 638512875, 172297125, 13956067125, 74246277105, 3093594879375, 14992036723125, 2143861251406875, 16436269594119375, 4226469324202125, 48028060502296875, 593531957565421875, 56437147443285984375

Offset: 0

N. J. A. Sloane, Jan 19 2017

This has the same start as two other sequences, A241591 and A248592, but appears to be different from both.

			Coefficients are 1, -4,-6, -154/3, -1610/3, -34588/5, -4666292/45, -553625626/315, -1158735422/35, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280780, A280781.

Cf. also A241591, A248592.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = {
  my(M = subst(x*Ser(A000699_seq(N)), x, x/(1-x)^2));
  Vec(x/(1-x)*exp(1-x/2-(1-x)^2/(2*x)*(2*M + M^2))/M);
};
apply(numerator, seq(18))  \\ Gheorghe Coserea, Jan 22 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

A376125 a(n) = 1 + Sum_{k=0..n-1} (2k+1) a(k) * a(n-k-1).

Original entry on oeis.org

1, 2, 9, 67, 681, 8556, 126253, 2124340, 39991633, 831271006, 18893178381, 465972248083, 12394713108433, 353750057246236, 10784915257548041, 349874160411051511, 12036066260440602401, 437714593034154481686, 16780944423208533034861, 676482338975579658794689

Offset: 0

Ilya Gutkovskiy, Sep 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..402

Cf. A000699, A007317, A321087.

a[n_] := a[n] = 1 + Sum[(2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 2 x^2 A'[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / ( (1 - x) *  (1 - x * A(x) - 2 * x^2 * A'(x)) ).
a(n) ~ c * 2^n * n * n!, where c = 0.6018110636400677977754542011395053310779724922160159... - Vaclav Kotesovec, Sep 11 2024
a(n) = 1 + n * Sum_{k=0..n-1} a(k) * a(n-1-k). - Seiichi Manyama, Jul 15 2025

A268814 Number of purely crossing partitions of [n].

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 5, 14, 62, 298, 1494, 8140, 47146, 289250, 1873304, 12756416, 91062073, 679616480, 5290206513, 42858740990, 360686972473, 3147670023632, 28439719809159, 265647698228954, 2561823514680235, 25475177517626196, 260922963832247729, 2749617210928715246

Offset: 0

Michel Marcus, Feb 14 2016

nonn

For the definition of a purely crossing partition refer to Dykema link (see PC(n) Definition 1.2 and Table 2).
From Gus Wiseman, Feb 23 2019: (Start)
For n >= 1, a set partition of {1,...,n} is purely crossing if it is topologically connected (A099947), has no successive elements in the same block (A000110(n - 1)), and the first and last vertices belong to different blocks (A005493(n - 2)). For example, the a(4) = 1, a(6) = 5, and a(7) = 14 purely crossing set partitions are:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}
(End)

			G.f.: A(x) = 1 + x^4 + 5*x^6 + 14*x^7 + 62*x^8 + 298*x^9 + 1494*x^10 + 8140*x^11 + 47146*x^12 +...

Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.

Cf. A000108 (non-crossing partitions), A000110, A000699, A001263, A002662, A005493, A016098, A054726, A099947, A268815, A306417, A324011, A324166, A324172, A324173, A324324.

n = 30; F = x*Sum[BellB[k] x^k, {k, 0, n}] + O[x]^n; B = ComposeSeries[1/( InverseSeries[F, w]/w)-1, x/(1+x) + O[x]^n]; A = (B-x)/(1+x); Join[{1}, CoefficientList[A, x] // Rest] (* Jean-François Alcover, Feb 23 2016, adapted from K. J. Dykema's code *)
intvQ[set_]:=Or[set=={},Sort[set]==Range[Min@@set,Max@@set]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And[!MatchQ[#,{_,{_,x_,y_,_},_}/;x+1==y],#=={}||And@@Not/@intvQ/@Union@@@Subsets[#,{1,Length[#]-1}],#=={}||Position[#,1][[1,1]]!=Position[#,n][[1,1]]]&]],{n,0,10}] (* Gus Wiseman, Feb 23 2019 *)

lista(nn) = {c = x/serreverse(x*serlaplace(exp(exp(x+x*O(x^nn)) -1))); b = subst(c, x, x/(1+x)+ O(x^nn)); vb = Vec(b-1); va = vector(#vb); va[1] = 0; va[2] = 0; for (k=3, #va, va[k] = vb[k] - va[k-1]; ); concat(1, va); }

{a(n) = my(A=1+x^3); for(i=1, n, A = sum(m=0, n, x^m/prod(k=1, m, (1+x)^2*A - k*x +x*O(x^n)) )/(1+x) ); polcoeff( A, n)}
for(n=0,35,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

{Stirling2(n, k) = n!*polcoeff(((exp(x+x*O(x^n)) - 1)^k)/k!, n)}
{Bell(n) = sum(k=0,n, Stirling2(n, k) )}
{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, Bell(m)*x^m/((1+x +x*O(x^n))^(2*m+1)*A^m)) ); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

G.f.: G(x) satisfies B(x) = x + (1 + x)*G(x) where B(x) is the g.f. of A268815 (see A(x) in Dykema link p. 7).
From Paul D. Hanna, Mar 07 2016: (Start)
O.g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} A000110(n)*x^n / ((1+x)^(2*n+1) * A(x)^n), where A000110 are the Bell numbers.
(2) A(x) = 1/(1+x) * Sum_{n>=0} x^n / Product_{k=1..n} ((1+x)^2*A(x) - k*x).
(3) A(x) = 1/(1+x - x/((1+x)*A(x) - 1*x/(1+x - x/((1+x)*A(x) - 2*x/(1+x - x/((1+x)*A(x) - 3*x/(1+x - x/((1+x)*A(x) - 4*x/(1+x - x/((1+x)*A(x) -...)))))))))), a continued fraction. (End)

A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 4, 3, 27, 31, 5, 248, 357, 117, 7, 2830, 4742, 2218, 314, 9, 38232, 71698, 42046, 9258, 690, 11, 593859, 1216251, 837639, 243987, 30057, 1329, 13, 10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15, 202601898, 472751962, 404979234, 166620434, 35456432, 3857904, 196532, 3812, 17, 4342263000, 10651493718, 9869474106, 4561150162, 1149976242, 160594860, 11946360, 426852, 5904, 19

Offset: 0

Gheorghe Coserea, May 21 2017

Row n>0 contains n terms.

"The series expansion of the solution counts skeleton vertex diagrams with dressed propagators and bare interactions." (see G^2v-skeleton expansion in Molinari link)

			A(x;t) = 1 + x + (4 + 3*t)*x^2 + (27 + 31*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]      [4]      [5]    [6]   [7]
[0]  1;
[1]  1;
[2]  4,        3;
[3]  27,       31,       5;
[4]  248,      357,      117,      7;
[5]  2830,     4742,     2218,     314,     9;
[6]  38232,    71698,    42046,    9258,    690,     11;
[7]  593859,   1216251,  837639,   243987,  30057,   1329,  13;
[8]  10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15;
[9] ...

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. A286781, A286782, A286783, A286784, A286785.

max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = ((1 + x*(1 + 2 t + x t^2) y0[x, t]^2 + t (1 - t)*x^2*y0[x, t]^3 + 2 x^2 y0[x, t] D[y0[x, t], x]))/(1 + 2 x*t) + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max - 1}] // Flatten (* Jean-François Alcover, May 23 2017, adapted from PARI *)

A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
concat(apply(p->Vecrev(p), Vec(A286795_ser(11))))
\\ test: y=A286795_ser(50); 0 == 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*y'

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 0 = 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*deriv(y,x), with y(0;t)=1, where P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.

A000699(n+1) = T(n,0), 1 = P_n(-1), A049464(n+1) = P_n(1).

cp's OEIS Frontend

A113672 Self-convolution 6th power equals A113666, where a(n) = n*A113666(n-1) for n>=1, with a(0)=1.

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A113673 Self-convolution 7th power equals A113667, where a(n) = n*A113667(n-1) for n>=1, with a(0)=1.

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A113674 Self-convolution 8th power equals A113668, where a(n) = n*A113668(n-1) for n>=1, with a(0)=1.

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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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Magma

Mathematica

SageMath

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A280778 Numerators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

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A280779 Denominators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

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

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Maple

Mathematica

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A376125 a(n) = 1 + Sum_{k=0..n-1} (2*k+1) * a(k) * a(n-k-1).

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A376125 a(n) = 1 + Sum_{k=0..n-1} (2k+1) a(k) * a(n-k-1).