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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

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

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

D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization ..., arXiv:hep-th/9912093, 1999, 2000.

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).

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.

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

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.

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).

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

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 problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.

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

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]. - 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

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David Broadhurst, Dec 14 1999

STATUS

approved

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Last modified September 2 00:17 EDT 2015. Contains 261279 sequences.