login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099398 Numerators of rationals (in lowest terms) used in a certain high temperature expansion. 4
1, 1, 1, 1, 7, 3, 33, 143, 143, 221, 4199, 2261, 7429, 37145, 334305, 570285, 1964315, 3411705, 23881935, 42077695, 149184555, 265937685, 3811773485, 6861192273, 24805848987, 135054066707, 327988447717, 599427163069, 6593698793759 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The rationals A(n) defined below appear in the expansion of the one-loop effective potential V1(y) for the thermal phi^4 model. See the Dolan-Jackiw, Kapusta and Quir/'os references. The expansion variable is y:=(m^2(phi))/(2*pi*k*T)^2 with Boltzmann's constant k, the (absolute) temperature T and m^2(phi):= m^2 + (lambda/2) phi^2 if the coupling constant is lambda/4! and the mass is m.

The relevant expansion of part of the thermal one-loop effective potential is ((pi^2)*((k*T)^4)/2)*sum(A(n)*Zeta(2*n+1)*(-1)^(n+1)*y^(n+2),n=1..infty) with Riemann's Zeta function. The expansion parameter y is given above. See the W. Lang link for more details.

REFERENCES

J. I. Kapusta, Finite-temperature field theory, Cambridge University Press, 1989.

M. Quirós, Field theory at finite temperature and phase transitions, Helv. Phys. Acta 67 (1994) 451-583.

LINKS

Table of n, a(n) for n=0..28.

Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys.Rev. D9,12 (1974) 3320-41.

Wolfdieter Lang, Rationals A(n) and more.

FORMULA

a(n) = numerator(A(n)) with A(n):= Catalan(n)/((n+2)*2^(2*n-1)) where Catalan(n):=A000108(n)=binomial(2*n, n)/(n+1).

a(n) = numerator(8*(2*n-1)!!/((2*(n+2))!!)) with the double factorials (2*n-1)!!:=A001147(n) (with (-1)!!:=1) and (2*n)!!:=A000165(n).

EXAMPLE

Rationals A(n):=A099398(n)/A099399(n), n>=0: 1/1, 1/6, 1/16, 1/32, 7/384, ...

CROSSREFS

The denominators are given in A099399.

Sequence in context: A290235 A225825 A199927 * A272276 A271597 A272505

Adjacent sequences: A099395 A099396 A099397 * A099399 A099400 A099401

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 10 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)