A000699 Number of irreducible chord diagrams with 2n nodes.

1, 1, 1, 4, 27, 248, 2830, 38232, 593859, 10401712, 202601898, 4342263000, 101551822350, 2573779506192, 70282204726396, 2057490936366320, 64291032462761955, 2136017303903513184, 75197869250518812754, 2796475872605709079512, 109549714522464120960474, 4509302910783496963256400, 194584224274515194731540740

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Perturbation expansion in quantum field theory: spinor case in 4 spacetime dimensions.

a(n)*2^(-n) is the coefficient of the x^(2*n-1) term in the series reversal of the asymptotic expansion of 2*DawsonF(x) = sqrt(Pi)*exp(-x^2)*erfi(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

A set partition is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y. Then a(n) is the number of topologically connected 2-uniform set partitions of {1...2n}. See my links for examples. - Gus Wiseman, Feb 23 2019

From Julien Courtiel, Oct 09 2024: (Start)

a(n) is the number of rooted bridgeless combinatorial maps with n edges (genus is not fixed). A map is bridgeless if it has no edge whose removal disconnects the graph. For example, for n=2, there are 4 bridgeless maps with 2 edges: 2 planar maps with 1 vertex (either two consecutive loops, or two nested loops), 1 toric map with 1 vertex, and 1 planar map with 2 vertices connected by a double edge.

Also, a(n) is the number of trees with n edges equipped with a binary tubing. A tube is a connected subgraph. A binary tubing of a tree is a nested set collection S of tubes such that 1. S contains the tube of all vertices 2. Every tube of S is either reduced to one vertex, or it can be can partitioned by 2 tubes of S.

(End)

Examples

			a(31)=627625976637472254550352492162870816129760 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 +...
where d/dx (A(x) - 1)^2/x = 1 + 4*x + 27*x^2 + 248*x^3 + 2830*x^4 +...
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..404 (terms up to n=100 from T. D. Noe)
• Serban T. Belinschi, Marek Bozejko, Franz Lehner, and Roland Speicher, The normal distribution is "boxplus"-infinitely divisible, arXiv:0910.4263 [math.OA], 2009-2010.
• Daniel J. Bernstein, S. Engels, T. Lange, R. Niederhagen, et al., Faster elliptic-curve discrete logarithms on FPGAs, Preprint, 2016.
• Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.
• Michael Borinsky, Generating asymptotics for factorially divergent sequences, El. J. Combin. 25 (2018) P4.1, Sect 7.1.
• Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 9.
• David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization ..., arXiv:hep-th/9912093, 1999, 2000.
• David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pad-Borel resummation, Phys. Lett. B 475 (2000), 63-70.
• Christian Brouder, On the trees of quantum fields, arXiv:hep-th/9906111, 1999, p. 6.
• Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
• Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011.
• Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
• Julien Courtiel, Karen Yeats, and Noam Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1611.04611 [math.CO], 2016.
• Serge Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
• Philippe Flajolet and Marc Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, pp. 191-201.
• Zhen Huang, Denis Golež, Hugo U. R. Strand, and Jason Kaye, Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting, arXiv:2503.19727 [cond-mat.str-el], 2025. See p. 21.
• Martin Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136.
• Martin Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
• Florian Kogelbauer and Ilya Karlin, On the Relation of Exact Hydrodynamics to the Chapman-Enskog Series, arXiv:2506.17441 [math-ph], 2025. See pp. 4-5.
• Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
• Ali Assem Mahmoud, An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams, arXiv:2009.12688 [math.CO], 2020.
• Ali Assem Mahmoud, Chord Diagrams and the Asymptotic Analysis of QED-type Theories, arXiv:2011.04291 [hep-th], 2020.
• Ali Assem Mahmoud, An asymptotic expansion for the number of two-connected chord diagrams, J. Math. Phys. (2023) Vol. 64, 122301. See Example 2.1.
• Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
• Nicolas Marie and Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, arXiv:1210.5457 [math.CO], 2012, Section 4.1.
• Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074).
• Vincent Pilaud and Juanjo Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440 [math.CO], 2013.
• J. Riordan, Letter, Jul 06 1978
• Einar A. Rødland, Pseudoknots in RNA Secondary Structures: Representation, Enumeration, and Prevalence, J. Comput. Biology, Vol 13, No 6 (2006), 1197-1213. (see equation 10)
• R. R. Stein, On a class of linked diagrams, I. Enumeration, J. Combin. Theory, A 24 (1978), 357-366.
• R. R. Stein and C. J. Everett, On a class of linked diagrams, II. Asymptotics, Discrete Math., 21 (1978), 309-318.
• Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
• Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259 (nonseparable integer systems on n pairs). (Give an incorrect a(6)=2720.)
• Gus Wiseman, The a(4) = 27 connected chord diagrams.
• Gus Wiseman, The a(5) = 248 connected chord diagrams.
• Gus Wiseman, Constructive Mathematica program for A000699.
• Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
• Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv:1803.10080 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
• Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.

Crossrefs

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A004300, A051862, A212273. Column sums of A232223. First column of A322402.

Cf. A007297, A016098, A099947, A136653, A268815, A306438, A324166, A324172, A324173, A324327.

Programs

Maple

A000699 := proc(n)
    option remember;
    if n <= 1 then
        1;
    else
        add((2*i-1)*procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc:
seq(A000699(n),n=0..30) ; # R. J. Mathar, Jun 12 2018

Mathematica

terms = 22; A[] = 0; Do[A[x] = x + x^2 * D[A[x]^2/x, x] + O[x]^(terms+1) // Normal, terms]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Apr 06 2012, after Paul D. Hanna, updated Jan 11 2018 *)
a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n]] = (n-1)*Sum[a[[i]]*a[[n-i]],{i,1,n-1}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 22 2014 *)
Module[{max = 20, s}, s = InverseSeries[ComplexExpand[Re[Series[2 DawsonF[x], {x, Infinity, 2 max + 1}]]]]; Table[SeriesCoefficient[s, 2 n - 1] 2^n, {n, 1, max}]] (* Vladimir Reshetnikov, Apr 23 2016 *)

PARI

{a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+x^2*deriv((A-1)^2/x)+x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]

PARI

{a(n) = my(A); A = 1+O(x) ; for( i=0, n, A = 1+x + (A-1)*(2*x*A' - A + 1)); polcoeff(A, n)}; /* Michael Somos, May 12 2012 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020] */

PARI

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(22)  \\ Gheorghe Coserea, Jan 22 2017

PARI

seq(n)={my(g=serlaplace(1 / sqrt(1 - 2*x + O(x*x^n)))); Vec(sqrt((x/serreverse( x*g^2 ))))} \\ Andrew Howroyd, Nov 21 2024

Python

def A000699_list(n):
    list = [1, 1] + [0] * (n - 1)
    for i in range(2, n + 1):
        list[i] = (i - 1) * sum(list[j] * list[i - j] for j in range(1, i))
    return list
print(A000699_list(22)) # M. Eren Kesim, Jun 23 2021

Formula

a(n) = (n-1)*Sum_{i=1..n-1} a(i)*a(n-i) for n > 1, with a(1) = a(0) = 1. [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]

A212273(n) = n * a(n). - Michael Somos, May 12 2012

G.f. satisfies: A(x) = 1 + x + x^2*[d/dx (A(x) - 1)^2/x]. - Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]

a(n) ~ n^n * 2^(n+1/2) / exp(n+1) * (1 - 31/(24*n) - 2207/(1152*n^2) - 3085547/(414720*n^3) - 1842851707/(39813120*n^4) - ...). - Vaclav Kotesovec, Feb 22 2014, extended Oct 23 2017

G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2*x/(A(x) - 3*x/(A(x) - 4*x/(A(x) - 5*x/(A(x) - ...))))), a continued fraction relation. - Paul D. Hanna, Nov 04 2020

G.f. A(x) satisfies: A(x*B(x)^2) = B(x) where B(x) is the g.f. of A001147. - Andrew Howroyd, Nov 21 2024

Extensions

More terms from David Broadhurst, Dec 14 1999

Inserted "chord" in definition. - N. J. A. Sloane, Jan 19 2017

Added a(0)=1. - N. J. A. Sloane, Nov 05 2020

Modified formulas slightly to include a(0) = 1. - Paul D. Hanna, Nov 06 2020