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A000699 Number of irreducible diagrams with 2n nodes.
(Formerly M3618 N1468)
24
1, 1, 4, 27, 248, 2830, 38232, 593859, 10401712, 202601898, 4342263000, 101551822350, 2573779506192, 70282204726396, 2057490936366320, 64291032462761955, 2136017303903513184, 75197869250518812754, 2796475872605709079512, 109549714522464120960474, 4509302910783496963256400, 194584224274515194731540740 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

REFERENCES

D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pad-Borel resummation. Phys. Lett. B 475 (2000), 63-70.

Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts), http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt.

Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011; http://jtburns.myweb.usf.edu/assembly/papers/Graphs_and_DNA_Recomb_2011.pdf.

M. 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.

M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.

Nijenhuis, Albert and Wilf, Herbert S., The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356-359.

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).

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.

J. Touchard, Sur un proble`me de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.

LINKS

T. D. Noe, Table of n, a(n) for n=1..100

D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization ...

Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926, 2011.

P. Flajolet and M. Noy, Analytic Combinatorics of Chord Diagrams

V. Pilaud, J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440, 2013

FORMULA

a(n) = (n-1)*Sum_{i=1..n-1} a(i)*a(n-i).

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

G.f. satisfies: A(x) = x + x^2*[d/dx A(x)^2/x]. [From Paul D. Hanna, Dec 31 2010]

a(n) ~ n^n * 2^(n+1/2) / exp(n+1). - Vaclav Kotesovec, Feb 22 2014

EXAMPLE

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) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 +...

where d/dx A(x)^2/x = 1 + 4*x + 27*x^2 + 248*x^3 + 2830*x^4 +...

MATHEMATICA

max = 18; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; coes = CoefficientList[ Series[f[x] - (x+x^2*D[f[x]^2/x, x]), {x, 0, max}], x]; sol = Solve[ Thread[coes == 0]]; a[n_] := c[n] /. sol[[1]]; Table[a[n], {n, 1, max}](* Jean-François Alcover, Apr 06 2012, after Paul D. Hanna *)

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 *)

PROG

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

(PARI) {a(n) = local(A); A = O(x) ; for( i=1, n, A = x + A * (2 * x * A' - A)); polcoeff(A, n)} /* Michael Somos, May 12 2012 */

CROSSREFS

Cf. A004300, A051862, A212273.

Sequence in context: A121063 A229619 A051863 * A138423 A239372 A239375

Adjacent sequences:  A000696 A000697 A000698 * A000700 A000701 A000702

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from David Broadhurst, Dec 14 1999

STATUS

approved

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Last modified October 31 17:56 EDT 2014. Contains 248868 sequences.