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A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.
(Formerly M1974 N0783)
43
1, 1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, 12277827850, 285764591114, 7213364729026, 196316804255522, 5731249477826890, 178676789473121834, 5925085744543837186, 208256802758892355202, 7734158085942678174730 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams. [Corrected by Charles R Greathouse IV, Jan 24 2014][Clarified by Robert Coquereaux, Sep 14 2014]

a(n+1) is the moment of order 2*n for the probability density function rho(x) = (1/sqrt(2*Pi))*exp(x^2/2)/[(u(x))^2+Pi/2], with u(x) = Integral_{t=0..x} exp(t*t/2) dt, on the real interval -infinity..infinity. - Groux Roland, Jan 13 2009

Starting (1, 2, 10, 74, ...) = INVERTi transform of A001147: (1, 3, 15, 105, ...). - Gary W. Adamson, Oct 21 2009

The Cvitanovic et al. paper relates this sequence to A005411 and A005413. - Robert Munafo, Jan 24 2010

Hankel transform of a(n+1) is A168467. - Paul Barry, Nov 26 2009

a(n) = number of labeled Dyck (n-1)-paths (A000108) in which each vertex that terminates an upstep is labeled with an integer i in [0,h], where h is the height of the vertex . For example UDUD contributes 4 labeled paths--0D0D, 0D1D, 1D0D, 1D1D where upsteps are replaced by their labels--and UUDD contributes 6 labeled paths to a(3)=10. The Deléham (Mar 24 2007) formula below counts these labeled paths by number of "0" labels. - David Callan, Aug 23 2011

a(n) is the number of indecomposable perfect matchings on [2n]. A perfect matching on [2n] is decomposable if a nonempty subset of the edges forms a perfect matching on [2k] for some k<n; otherwise it is indecomposable. For example, the perfect matching 1-2,3-4 is decomposable, and a(2) = 2 counts 1-3,2-4 and 1-4,2-3. - David Callan, Nov 29 2012

From Robert Coquereaux, Sep 12 2014: (Start)

QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff. They may have internal and external (i.e., pendant) lines. The order is the number of (internal) vertices. Vanishing diagrams: QED diagrams containing loops of type f with an odd number of vertices are set to 0 (Furry theorem). Proper diagrams: connected QED diagrams that remain connected when an arbitrary internal line is cut.

The number of Feynman diagrams of order 2n for the electron propagator (2-point function of QED), vanishing or not, proper or not, of order 2n, starting from n = 0, is given by 1, 2, 10, 74, 706, 8162, ..., i.e., this sequence A000698, with the first term (equal to 1) dropped. Call Sf the associated g.f.

The number of non-vanishing Feynman diagrams, for the same 2-point function, is given by 1, 1, 4, 25, 208, 2146, ..., i.e., by the sequence A005411, with a first term of order 0, equal to 1, added. Call S the associated g.f.

If one does not remove the vanishing diagram, but, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams (vanishing and non-vanishing) for the self-energy function of QED, 0, 1, 3, 21, 207, 2529, ..., i.e., the sequence A115974 with a first term of order 0, equal to 0, added. A115974 is twice A167872. Call Sigmaf the associated g.f.

If one removes the vanishing diagrams and, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams for the self-energy function of QED given by 0, 1, 3, 18, 153, 1638, ..., i.e., by the sequence A005412, with a first term of order 0, equal to 0, added. Call Sigma the associated g.f.

Then Sf = 1/(1-Sigmaf) and S = 1/(1-Sigma). (End)

For n>0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015

Also, counts certain isomorphism classes of closed normal linear lambda terms. [N. Zeilberger, 2015]. - N. J. A. Sloane, Sep 18 2016

REFERENCES

Dubois C., Giorgetti A., Genestier R. (2016) Tests and Proofs for Enumerative Combinatorics. In: Aichernig B., Furia C. (eds) Tests and Proofs. TAP 2016. Lecture Notes in Computer Science, vol 9762. Springer.

R. W. Robinson, Counting irreducible Feynman diagrams exactly and asymptotically, Abstracts Amer. Math. Soc., 2002, #975-05-270.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12.

F. Battaglia, T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247

S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, arXiv preprint arXiv:1111.4710 [math.CO], 2011.

S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, Discrete Math. 313 (2013), no. 3, 258-268. MR2995390.

Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts)

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

Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011.

P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.34 and fig 2b).

M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.

Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv:1504.06904 [math.SP], 2015.

G. Edgar, Transseries for beginners,  arXiv:0801.4877v5 [math.RA], 2008-2009.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.

R. J. Mathar, Table of Feynman Diagrams of the Interacting Fermion Green's Function, Int. J. Quant. Chem. 107 (2007) 1975. Also here.

L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, Phys. Rev. B, 71 (2005), 113102.

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

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

Wikipedia, Feynman diagram

Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003.

FORMULA

G.f.: 2 - 1/(1 + Sum_{n>=1} (2*n-1)!! * x^n ).

a(n+1) = Sum_{k=0..n} A089949(n, k)*2^k. - Philippe Deléham, Aug 15 2005

a(n+1) = Sum_{k=0..n} A053979(n,k). - Philippe Deléham, Mar 24 2007

From Paul Barry, Nov 26 2009: (Start)

G.f.: 1+x/(1-2x/(1-3x/(1-4x/(1-5x/(1-6x/(1-... (continued fraction).

G.f.: 1+x/(1-2x-6x^2/(1-7x-20x^2/(1-11x-42x^2/(1-15x-72x^2/(1-19x-110x^2/(1-... (continued fraction). (End)

G.f.: 1 + x * B(x) * C(x) where B(x) is the g.f. for A001147 and C(x) is the g.f. for A005416. - Michael Somos, Feb 08 2011

G.f.: 1+x/W(0); where W(k)=1+x+x*2k-x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011

Recurrence relation: a(n+1) = (2*n-1)*a(n) + sum {k = 1..n} a(k)*a(n+1-k) for n >= 0 and a(1) = 1.

The o.g.f. B(x) = sum {n>=1} a(n)*x^(2*n-1) = x + 2*x^3 + 10*x^5 + 74*x^7 + ... satisfies the Riccati differential equation y'(x) = -1/x^2 + 1/x^3*y(x) - 1/x^2*y(x)^2 with initial condition y(0) = 0 (Cf. A005412). The solution is B(x) = 1/z(x) + 1/x , where z(x) = - sum {n>=0} A001147(n) * x^(2*n+1) = -(x + x^3 + 3*x^5 + 15*x^7 +...). The function b(x) = - B(1/x) satisfies b'(x) = -1 - (x + b(x))*b(x). Hence the differential operator (D^2 + x*D + 1), where D = d/dx, factorizes as (D - a(x))*(D - b(x)), where a(x) = -(x + b(x)), as conjectured by [Edgar, Problem 4.32]. For a refinement of this sequence see A053979. - Peter Bala, Dec 22 2011

From Sergei N. Gladkovskii, Aug 19 2012, Oct 24 2012, Mar 19 2013, May 20 2013, May 29 2013, Aug 04 2013, Aug 05 2013: (Start) Continued Fractions:

G.f.: 2-G(0) where G(k)= 1 -(k+1)*x/G(k+1).

G.f.: 2 - U(0) where U(k)= 1 - (2*k+1)*x/(1 - (2*k+2)*x/U(k+1)).

G.f.: 2 - U(0) where U(k)= 1 - (4*k+1)*x - (2*k+1)*(2*k+2)*x^2/U(k+1)).

G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+2)/(1 - x*(2*k+3)/Q(k+1)).

G.f.: 1 + x/Q(0) where Q(k)= 1 - x*(k+2)/Q(k+1)).

G.f.: 2 - G(0)/2 where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/ G(k+1))).

G.f.: 1 + x*G(0) where G(k)= 1 - x*(k+2)/(x*(k+2) - 1/G(k+1)).

G.f.: 2 - 1/B(x) where B(x) is the g.f. of A001147.

G.f.: 1 + x/(1-2*x*B(x)) where B(x) is the g.f. of A167872. (End)

a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 10 2014

G.f.: 1/x+(E^((1/2)/x) Sqrt[2/\[Pi]] Sqrt[-(1/x)])/Erfc[Sqrt[-(1/x)]/Sqrt[2] where Erfc(z)=1-Erf(z) is the complementary error function, and Erf(z) is the integral of the Gaussian distribution. This generating function is obtained from the generating functional of (4-dimensional) QED, evaluated in dimension 0 for the 2-point function, without the modification implementing Furry theorem. - Robert Coquereaux, Sep 14 2014

From Peter Bala, May 23 2017: (Start)

G.f. A(x) = 1 + x/(1 + x - 3*x/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - ... )))).

A(x) = 1 + x/(1 + x - 3*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - ... ))))))). (End)

EXAMPLE

G.f. = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 706*x^5 + 8162*x^6 + 110410*x^7 + ...

MAPLE

A006882 := proc(n) option remember; if n <= 1 then 1 else n*procname(n-2); fi; end;

A000698:=proc(n) option remember; global df; local k; if n=0 then RETURN(1); fi; A006882(2*n-1) - add(A006882(2*k-1)*A000698(n-k), k=1..n-1); end;

A000698 := proc(n::integer) local resul, fac, pows, c, c1, p, i ; if n = 0 then RETURN(1) ; else pows := combinat[partition](n) ; resul := 0 ; for p from 1 to nops(pows) do c := combinat[permute](op(p, pows)) ; c1 := op(1, c) ; fac := nops(c) ; for i from 1 to nops(c1) do fac := fac*doublefactorial(2*op(i, c1)-1) ; od ; resul := resul-(-1)^nops(c1)*fac ; od : fi ; RETURN(resul) ; end; # R. J. Mathar, Apr 24 2006

# alternative Maple program:

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,

      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +

                   b(x-1, y+1, true)  ))

    end:

a:= n-> `if`(n=0, 1, b(2*n-2, 0, false)):

seq(a(n), n=0..25);  # Alois P. Heinz, May 23 2015

a_list := proc(len) local n, A; if len=1 then return [1] fi: A := Array(-1..len-2); A[-1] := 1; A[0] := 1; for n to len-2 do A[n] := (2*n-1)*A[n-1]+add(A[j]*A[n-j-1], j=0..n-1) od: convert(A, list) end: a_list(20); # Peter Luschny, Jul 18 2017

MATHEMATICA

a[n_] := a[n] = (2n - 1)!! - Sum[ a[n - k](2k - 1)!!, {k, n-1}]; Array[a, 18, 0] (* Ignacio D. Peixoto, Jun 23 2006 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 2 - 1 / Sum[ (2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Nov 16 2011 *)

a[n_]:= SeriesCoefficient[1/x+(E^((1/2)/x) Sqrt[2/\[Pi]] Sqrt[-(1/x)])/Erfc[Sqrt[-(1/x)]/Sqrt[2]], {x, 0, n}, Assumptions -> x >0](* Robert Coquereaux, Sep 14 2014 *)

max = 20; g = t/Fold[1 - ((t + #2)*z)/#1 &, 1, Range[max, 1, -1]]; T[n_, k_] := SeriesCoefficient[g, {z, 0, n}, {t, 0, k}]; a[0] = 1; a[n_] := Sum[T[n-1, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Philippe Deléham *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 - 1 / sum( k=0, n, x^k * (2*k)! /(2^k * k!), x * O(x^n)), n))}; /* Michael Somos, Feb 08 2011 */

(PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

(Python)

from sympy import factorial2

class Memoize:

    def __init__(self, func):

        self.func = func

        self.cache = {}

    def __call__(self, arg):

        if arg not in self.cache:

            self.cache[arg] = self.func(arg)

        return self.cache[arg]

@Memoize

def a(n): return 1 if n==0 else factorial2(2*n - 1) - sum([factorial2(2*k - 1)*a(n - k) for k in xrange(1, n)])

print map(a, xrange(101)) # Indranil Ghosh, Jul 18 2017

CROSSREFS

Cf. A004208, A000165, A001147, A005412, A053979, A005411, A115974, A167872.

Column k=1 of A258219, A258222.

Sequence in context: A152408 A046863 A185971 * A289313 A092881 A004123

Adjacent sequences:  A000695 A000696 A000697 * A000699 A000700 A000701

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Richard Ehrenborg, Gabor Hetyei

EXTENSIONS

Formula corrected by Ignacio D. Peixoto, Jun 23 2006

More terms from Sean A. Irvine, Feb 27 2011

STATUS

approved

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Last modified August 21 02:25 EDT 2017. Contains 290855 sequences.