0,3

Also number of nonisomorphic unlabeled connected Feynman diagrams of order n.

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)=int(exp(t*t/2),t=0..x), on the real interval -infinity..infinity [From Groux Roland (roland.groux(AT)orange.fr), Jan 13 2009]

Starting (1, 2, 10, 74,...) = INVERTi transform of A0001147: (1, 3, 15, 105,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces..., Discrete Math., 215 (2000), 1-12.

P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.

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

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

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

P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).

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

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links

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

a(n+1) = Sum_{k, 0<=k<=n} A089949(n, k)*2^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 15 2005

a(n+1)=Sum_{k, 0<=k<=n}A053979(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 24 2007

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

A000698:=proc(n) option remember; global df; local k; if n=0 then RETURN(1); fi; df(2*n-1) - add(df(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: for n from 0 to 20 do printf("%a, ", A000698(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2006

a[ 0 ] = 1; a[ n_ ] := a[ n ] = (2n - 1)!! - Sum[ a[ n - k ](2k - 1)!!, {k, 1, n-1} ] {#, a[ # ]} & /@ Range[ 17 ] // TableForm - Ignacio D. Peixoto, Jun 23 2006

Cf. A004208, A000165.

Cf. A0001147 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]

Sequence in context: A141149 A152408 A046863 this_sequence A092881 A004123 A086352

Adjacent sequences: A000695 A000696 A000697 this_sequence A000699 A000700 A000701

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), ehrenbor(AT)catalan.math.uqam.ca (Richard Ehrenborg), hetyei(AT)lacim.uqam.ca (G. Hetyei).

Formula corrected by Ignacio D. Peixoto, Jun 23 2006