Search: id:A000698
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%I A000698 M1974 N0783
%S A000698 1,1,2,10,74,706,8162,110410,1708394,29752066,576037442,12277827850,
%T A000698 285764591114,7213364729026,196316804255522,5731249477826890,
%U A000698 178676789473121834,5925085744543837186
%N 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!.
%C A000698 Also number of nonisomorphic unlabeled connected Feynman diagrams of 
               order n.
%C A000698 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]
%C A000698 Starting (1, 2, 10, 74,...) = INVERTi transform of A0001147: (1, 3, 15, 
               105,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]
%D A000698 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000698 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000698 D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces..., Discrete 
               Math., 215 (2000), 1-12.
%D A000698 P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman 
               diagrams, Phys. Rev. D 18 (1978), 1939-1949.
%D A000698 L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, 
               Phys. Rev. B 71 (2005), 113102.
%D A000698 R. W. Robinson, Counting irreducible Feynman diagrams exactly and asymptotically, 
               Abstracts Amer. Math. Soc., 2002, #975-05-270.
%D A000698 J. Touchard, Sur un proble`me de configurations et sur les fractions 
               continues, Canad. J. Math., 4 (1952), 2-25.
%H A000698 P. Cvitanovic, B. Lautrup and R. B. Pearson, <a href="http://www.nbi.dk/
               ~predrag/papers/PRD18-78.pdf">The number and weights of Feynman diagrams</
               a>, Phys. Rev. D18, 1939 (1978).
%H A000698 L. G. Molinari, <a href="http://dx.doi.org/10.1103/PhysRevB.71.113102">
               Hedin's equations and enumeration of Feynman diagrams</a>, Phys. 
               Rev. B, 71 (2005), 113102.
%H A000698 P. Zinn-Justin and J.-B. Zuber, <a href="http://arXiv.org/abs/math-ph/
               0303049">Matrix integrals and the generation and counting of virtual 
               tangles and links</a>
%F A000698 G.f.: 1 - 1/{1 + sum_{n >= 1} (2n-1)!! * x^n}.
%F A000698 a(n+1) = Sum_{k, 0<=k<=n} A089949(n, k)*2^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 15 2005
%F A000698 a(n+1)=Sum_{k, 0<=k<=n}A053979(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 24 2007
%p A000698 df := proc(n) option remember; if n <= 1 then 1 else n*df(n-2); fi; end;
%p A000698 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;
%p A000698 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
%t A000698 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
%Y A000698 Cf. A004208, A000165.
%Y A000698 Cf. A0001147 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]
%Y A000698 Sequence in context: A141149 A152408 A046863 this_sequence A092881 A004123 
               A086352
%Y A000698 Adjacent sequences: A000695 A000696 A000697 this_sequence A000699 A000700 
               A000701
%K A000698 nonn,easy,nice
%O A000698 0,3
%A A000698 N. J. A. Sloane (njas(AT)research.att.com), ehrenbor(AT)catalan.math.uqam.ca 
               (Richard Ehrenborg), hetyei(AT)lacim.uqam.ca (G. Hetyei).
%E A000698 Formula corrected by Ignacio D. Peixoto, Jun 23 2006

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