|
Search: id:A000698
|
|
|
| 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)
|
|
+0
14
|
|
| 1, 1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, 12277827850, 285764591114, 7213364729026, 196316804255522, 5731249477826890, 178676789473121834, 5925085744543837186
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Also number of nonisomorphic unlabeled connected Feynman diagrams of order n.
|
|
REFERENCES
|
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.
|
|
LINKS
|
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
|
|
FORMULA
|
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
|
|
MAPLE
|
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
|
|
MATHEMATICA
|
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
|
|
CROSSREFS
|
Cf. A004208, A000165.
Adjacent sequences: A000695 A000696 A000697 this_sequence A000699 A000700 A000701
Sequence in context: A047853 A141149 A046863 this_sequence A092881 A004123 A086352
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
njas, ehrenbor(AT)catalan.math.uqam.ca (Richard Ehrenborg), hetyei(AT)lacim.uqam.ca (G. Hetyei).
|
|
EXTENSIONS
|
Formula corrected by Ignacio D. Peixoto, Jun 23 2006
|
|
|
Search completed in 0.002 seconds
|
