|
Search: id:A115974
|
|
|
| A115974 |
|
Number of Feynman diagrams of the proper self-energy at perturbative order n. The number of diagrams of A000698 left if the singly connected improper diagrams are removed: a(n)<=A000698(n+1). G.f. is essentially the inversion of the G.f. of A000698. |
|
+0
3
|
|
| 2, 6, 42, 414, 5058, 72486, 1182762, 21573054, 434358018, 9565348806, 228740050602, 5904853053534, 163728751178178, 4855046674314726, 153367360732387242, 5143219420761900414, 182530741698302811138, 6835913695777897799046, 269455018264860747728682, 11152465473005099074500894, 483617145128737549802831298
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
REFERENCES
|
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971
|
|
LINKS
|
R. J. Mathar, A table of Third and Fourth Order Feynman Diagrams of the Interacting Fermion Green's Function, arXiv:physics/0512022 (2005)
|
|
FORMULA
|
a(n)=A000698(n+1)-Sum(m=1..n-1)*a(m)*A000698(n+1-m); 1-Sum(n=1..infinity)a(n)*x^n =1/[1+Sum(n=1..infinity)A000698(n+1)*x^n] (G.f.)
|
|
EXAMPLE
|
There are A000698(3)=10 self-energy diagrams of order n=2. Four of them are
chained diagrams of order n=1 (of two kinds) which are simply connected, which
leaves 10-4=6=a(2) proper diagrams.
|
|
MAPLE
|
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: A115974 := proc(n::integer) local resul, m ; resul := A000698(n+1) ; for m from 1 to n-1 do resul := resul-A115974(m)*A000698(n+1-m) ; od: RETURN(resul) ; end: for n from 1 to 20 do printf("%a, ", A115974(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2006
|
|
CROSSREFS
|
Cf. A000698.
Sequence in context: A074015 A074021 A050862 this_sequence A066864 A116896 A061062
Adjacent sequences: A115971 A115972 A115973 this_sequence A115975 A115976 A115977
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 15 2006
|
|
EXTENSIONS
|
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2006, Nov 07 2006
|
|
|
Search completed in 0.002 seconds
|
