A000699 -id:A000699 - cp's OEIS frontend

A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.

1, 1, 2, 4, 8, 27, 354

Offset: 0

Gus Wiseman, Feb 22 2019

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

			The a(0) = 1 through a(5) = 27 graphs:
  {}  {}  {}      {}      {}          {}
          {{12}}  {{12}}  {{12}}      {{12}}
                  {{13}}  {{13}}      {{13}}
                  {{23}}  {{14}}      {{14}}
                          {{23}}      {{15}}
                          {{24}}      {{23}}
                          {{34}}      {{24}}
                          {{13}{24}}  {{25}}
                                      {{34}}
                                      {{35}}
                                      {{45}}
                                      {{13}{24}}
                                      {{13}{25}}
                                      {{14}{25}}
                                      {{14}{35}}
                                      {{24}{35}}
                                      {{13}{14}{25}}
                                      {{13}{24}{25}}
                                      {{13}{24}{35}}
                                      {{14}{24}{35}}
                                      {{14}{25}{35}}
                                      {{13}{14}{24}{25}}
                                      {{13}{14}{24}{35}}
                                      {{13}{14}{25}{35}}
                                      {{13}{24}{25}{35}}
                                      {{14}{24}{25}{35}}
                                      {{13}{14}{24}{25}{35}}

Cf. A000108, A000699, A001764, A002061, A007297, A016098, A054726 (non-crossing chord graphs), A099947, A136653, A268814.

Cf. A324168, A324169, A324172, A324173, A324323, A324327 (covering case).

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[crosscmpts[#]]<=1&]],{n,0,5}]

Binomial transform of A324327.

A111088 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 2.

Original entry on oeis.org

1, 1, 2, 8, 52, 464, 5184, 68928, 1057584, 18345536, 354570112, 7551674624, 175700025728, 4433961734656, 120642462777344, 3520972469815296, 109731998026937088, 3637456413350962176, 127800512612435896320

Offset: 0

Philippe Deléham, Oct 10 2005

nonn

For x = 1, this is : 1, 1, 1, 2, 7, 34, 206, 1476, 12123, ..., see A075834.
For x = 0, this is : 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
For x = -1, this is : 1, 1, -1, 2, -5, 14, -42, 132, -429, ...,((-1)^(n+1)* A000108(n)).
a(n)*2^(-n) is the coefficient at the x^(n-1) term in the series reversal of the asymptotic expansion of 2 * DawsonF(sqrt(x))/sqrt(x) = sqrt(Pi) * exp(-x) * erfi(sqrt(x)) / sqrt(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016

			From _Paul D. Hanna_, Aug 02 2014: (Start)
O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 52*x^4 + 464*x^5 + 5184*x^6 +...
where A(x) = x/Series_Reversion(x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 945*x^6 +...)
and thus
A(x) = 1 + x/A(x) + 3*x^2/A(x)^2 + 15*x^3/A(x)^3 + 105*x^4/A(x)^4 + 945*x^5/A(x)^5 +...
Illustration of the initial terms:
a(2) = 2;
a(3) = 2*2^2 = 8;
a(4) = 2*3*8 + 1*2*2 = 52;
a(5) = 2*4*52 + 1*2*8 + 2*8*2 = 464;
a(6) = 2*5*464 + 1*2*52 + 2*8*8 + 3*52*2 = 5184; ...
To illustrate formula: [x^(n+1)] A(x)^n = 2*n*([x^n] A(x)^n), form a table of coefficients of x^k in A(x)^n:
n=1: [1, 1,  2,   8,   52,   464,  5184,   68928,  1057584, ...];
n=2: [1, 2,  5,  20,  124,  1064, 11568,  150912,  2283888, ...];
n=3: [1, 3,  9,  37,  222,  1836, 19412,  248256,  3703536, ...];
n=4: [1, 4, 14,  60,  353,  2824, 29032,  363696,  5345040, ...];
n=5: [1, 5, 20,  90,  525,  4081, 40810,  500480,  7241460, ...];
n=6: [1, 6, 27, 128,  747,  5670, 55205,  662460,  9431172, ...];
n=7: [1, 7, 35, 175, 1029,  7665, 72765,  854197, 11958758, ...];
n=8: [1, 8, 44, 232, 1382, 10152, 94140, 1081080, 14876033, ...]; ...
then we can see that the diagonals are related in the following way:
[2, 20, 222, 2824, 40810, 662460, 11958758, ...]
= [2*1, 4*5, 6*37, 8*353, 10*4081, 12*55205, 14*854197, ...].
Also, the diagonal
[1, 5, 37, 353, 4081, 55205, 854197, 14876033, ...]
is the logarithmic derivative of the g.f. of the double factorials.
Further, the main diagonal in the above table equals:
[1, 2*1, 3*3, 4*15, 5*105, 6*945, 7*10395, 8*135135, ...].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Alman, C. Lian, B. Tran, Circular Planar Electrical Networks: Posets and Positivity, 2013.

Cf. A001147, A232967, A000699.

x = 2; a[0] = a[1] = 1; a[2] = x; a[3] = 2x^2; a[n_] := a[n] = x*(n - 1)*a[n - 1] + Sum[(j - 1)*a[j]*a[n - j], {j, 2, n - 2}]; Table[ a[n], {n, 0, 18}] (* Robert G. Wilson v *)
Module[{max = 20, s}, s = InverseSeries[Series[2 DawsonF[Sqrt[x]]/Sqrt[x], {x, Infinity, max + 1}][[2, 2, 2]]]; Table[SeriesCoefficient[s, n-1] 2^n, {n, 0, max}]] (* Vladimir Reshetnikov, Apr 23 2016 *)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,(2*(k-1))!/(k-1)!/2^(k-1)))))[n+1] /* Paul D. Hanna, Jul 09 2006 */

/* From o.g.f. A = 1 + x*(A + x*A')/(A - x*A'): */
{a(n)=local(A=1+x); for(i=1, n, A=1 + x*(A+x*A')/(A-x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

O.g.f. A(x) satisfies:
(1) A(x) = x / Series_Reversion(x*G(x)) where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) such that G(x) is the g.f. of the double factorials (A001147). - Paul D. Hanna, Jul 09 2006
(2) A(x) = Sum_{n>=0} A001147(n) * x^n / A(x)^n, where A001147(n) = (2*n)!/(n!*2^n). - Paul D. Hanna, Aug 02 2014
(3) A(x) = 1 + x * (A(x) + x*A'(x)) / (A(x) - x*A'(x)). - Paul D. Hanna, Aug 02 2014
(4) [x^(n+1)] A(x)^n = 2*n*([x^n] A(x)^n) for n>=0. - Paul D. Hanna, Aug 02 2014
a(n) ~ 2^(n+1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Aug 02 2014

More terms from Robert G. Wilson v, Oct 12 2005

A113129 Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 6, 0, 0, 0, 10, 24, 0, 0, 0, 4, 82, 120, 0, 0, 0, 0, 84, 672, 720, 0, 0, 0, 0, 27, 1236, 5820, 5040, 0, 0, 0, 0, 0, 930, 16328, 54288, 40320, 0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880, 0, 0, 0, 0, 0, 0, 12452, 396528, 2775432

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Let R(m,n,k), 0<=k<=n, the Riordan array (1,x*g(x)) where g(x) is g.f. of the m-fold factorials . Then R(m,n,k) = R(m,n-1,k-1) + Sum_{j, 0<=j<=n-1-k} R(m,n-1,k+j)*P_m(j), R(m,n,0) = 0^n and R(m,0,k) = 0 if k>n.

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 6;
.0, 0, 0, 10, 24;
.0, 0, 0, 4, 82, 120;
.0, 0, 0, 0, 84, 672, 720;
.0, 0, 0, 0, 27, 1236, 5820, 5040;
.0, 0, 0, 0, 0, 930, 16328, 54288, 40320;
.0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880;
.0, 0, 0, 0, 0, 0, 12452, 396528, 2775432, 6003360, 362880;
.0, 0, 0, 0, 0, 0, 2830, 38732, 7057308, 37831752, 71019360, 39916800;

Cf. A000007, A000108, A000142, A000699.

R(m, n, k) : A097805 (m=0), A084938 (m=1), A111106 (m=2), A113333 (column sums).

P_0(x) = 1, P_1(x) = x, P_2(x) = 2*x^2, P_ n(x) = n*x*P_(n-1)(x) + Sum_{j, 1<=j<=n-1} j*P_j(x)*P_(n-1-j)(x).
P_n(x) = Sum_{k, 0<=k<=n} T(n, k)*x^k.
P_n(0) = A000007(n).
P_n(x) = A075834(n+1), A111088(n+1), A113130(n+1), A113131(n+1), A113132(n+1), A113133(n+1), A113134(n+1), A113135(n+1) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
P_n(-1) = (-1)^n*A000108(n), signed Catalan numbers.
T(n, n) = n! = A000142(n).
T(2*n+1, n+1) = A000699(n+1) (number of irreducible diagrams with 2n+2 nodes).
T(2*n+2, n+2) = A113332(n) = A000699(n+2)*(2*n+3)*(n+2)/(3*(n+1)).

Corrected by Philippe Deléham, Dec 18 2008

A113662 G.f. satisfies: A(x) = (1 + x(d/dx xA(x)) )^2.

Original entry on oeis.org

1, 2, 9, 62, 566, 6372, 84837, 1300214, 22511322, 434226300, 9231983850, 214481625516, 5406323440492, 146963638311880, 4286068830850797, 133501081493969574, 4423404073559930162, 155359770700317171084

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Self-convolution of A000699 (after ignoring the initial term), [previous name].

			G.f. A(x) = 1 + 2*x + 9*x^2 + 62*x^3 + 566*x^4 + 6372*x^5 + 84837*x^6 + 1300214*x^7 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000699, A113663, A113664, A113665, A113666, A113667, A113668.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=(1+x*deriv(x*A))^2);polcoeff(A,n,x)}

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;
};
Vec(sqr(Ser(A000699_seq(N))))  \\ Gheorghe Coserea, Jan 23 2017

G.f. satisfies: A(x) = (1 + x*(d/dx x*A(x)) )^2.

a(n) ~ 2^(n + 5/2) * n^(n+1) / exp(n+1). - Vaclav Kotesovec, Oct 23 2020

Name replaced with an existing formula by Paul D. Hanna, Sep 16 2024

A113669 Self-convolution cube equals A113663, where a(n) = n*A113663(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 6, 63, 904, 16080, 337374, 8107743, 218940480, 6554205342, 215319184860, 7701064928370, 297912862462680, 12396725926132990, 552257670588677214, 26229243983909050215, 1323230977463353055616, 70673562984581535191094

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A113663, A000699, A113670, A113671, A113672, A113673, A113674.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^3));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^3],
(2) [x^n] exp( x*A(x)^3 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^3 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018
From Vaclav Kotesovec, Oct 23 2020: (Start)
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.2509528330393045762351289...
a(n) ~ A113663(n)/3. (End)
a(0) = 1; a(n) = n * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 25 2021

A280775 Number of monolithic chord diagrams with n chords.

Original entry on oeis.org

1, 3, 11, 65, 573, 6547, 89639, 1414417, 25148617, 496416579, 10762275539, 254153371121, 6494217863461, 178558132802259, 5257524611172751, 165089697983580641, 5507950426778674129, 194605351254360182403, 7259714571747394749147, 285174902634083710549601

Offset: 1

N. J. A. Sloane, Jan 19 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..200
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016. See M_n.

Cf. A000699.

terms = 20;
c[_] = 0;
Do[c[x_] = x + x^2*D[c[x]^2/x, x] + O[x]^(terms+1) // Normal, terms];
c[x/(1-x)^2] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 01 2018 *)

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;
};
N = 20; Vec(subst(x*Ser(A000699_seq(N)), x, x/(1-x)^2))  \\ Gheorghe Coserea, Jan 22 2017

G.f.: C(x/(1-x)^2), where C(x) is the g.f. for A000699.

A280780 Numerators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

Original entry on oeis.org

1, -4, 2, -40, -182, -7624, -202652, -14115088, -30800534, -16435427656, -1051314228316, -22675483971248, -6980651581556876, -283099764343781072, -163910651754113166328, -43009695328217994139936, -793529010007812171331166, -20144221762701827321778088, -274475989492312981198559876

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -4, 2, -40/3, -182/3, -7624/15, -202652/45, -14115088/315, -30800534/63, -16435427656/2835, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..100
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280779, A280781.

seq(N) = {
  my(f = serreverse(x*Ser(vector(N, n, n!))));
  Vec(x* f'/f * exp(2 + (f-x)/(x*f)));
};
apply(numerator, seq(20))  \\ Gheorghe Coserea, Jan 22 2017

A111111(n) ~ n!*exp(-2)*(1 - 4/n + 2/(n*(n-1)) - (40/3)/(n*(n-1)*(n-2)) - ...). - Gheorghe Coserea, Jan 23 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280781 Denominators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 63, 2835, 14175, 22275, 467775, 1216215, 42567525, 638512875, 638512875, 834978375, 558242685, 1856156927625, 713906510625, 17717861581875, 2143861251406875, 9861761756471625, 147926426347074375, 75472666503609375, 48076088562799171875

Offset: 0

N. J. A. Sloane, Jan 19 2017

Has the same start as A046983 but is a different sequence.

			Coefficients are 1, -4, 2, -40/3, -182/3, -7624/15, -202652/45, -14115088/315, -30800534/63, -16435427656/2835, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..100
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280779, A280780.

Cf. also A046983.

seq(N) = {
  my(f = serreverse(x*Ser(vector(N, n, n!))));
  Vec(x* f'/f * exp(2 + (f-x)/(x*f)));
};
apply(denominator, seq(28))  \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A113670 Self-convolution 4th power equals A113664, where a(n) = n*A113664(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 8, 114, 2224, 53725, 1528200, 49703108, 1813503712, 73247619060, 3242579748000, 156107189374202, 8121266448765936, 454110696002834806, 27165980379205109232, 1731608155057922555400, 117183510733473232477120

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Cf. A113664, A000699, A113669, A113671, A113672, A113673, A113674.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^4));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^4],
(2) [x^n] exp( x*A(x)^4 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^4 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113671 Self-convolution 5th power equals A113665, where a(n) = n*A113665(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 10, 180, 4440, 135525, 4866156, 199577910, 9174096960, 466435229220, 25973117225450, 1571873641094680, 102741164109622800, 7214517196021315830, 541781124945022815720, 43336510897320779553450

Offset: 0

Paul D. Hanna, Nov 04 2005

nonn

Cf. A113665, A000699, A113669, A113670, A113672, A113673, A113674.

{a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^5));polcoeff(A,n,x)}

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^5],
(2) [x^n] exp( x*A(x)^5 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^5 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

cp's OEIS Frontend

A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.

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A111088 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 2.

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A113129 Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

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A113662 G.f. satisfies: A(x) = (1 + x*(d/dx x*A(x)) )^2.

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A113669 Self-convolution cube equals A113663, where a(n) = n*A113663(n-1) for n>=1, with a(0)=1.

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A280775 Number of monolithic chord diagrams with n chords.

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Mathematica

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A280780 Numerators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

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A280781 Denominators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

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A111088 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 2.

A113662 G.f. satisfies: A(x) = (1 + x(d/dx xA(x)) )^2.